Calculating Speed and Distance

Oct 13, 2025 by windailys.com 66 views

Introduction

When it comes to road trips, understanding speed and distance is crucial for planning and navigation. In this article, we will explore the concept of speed and distance in the context of Lina's road trip. We will analyze the given information and determine the correct expressions that represent Lina's speed during the first part of her trip.

Problem Statement

For the first 80 miles of her road trip, Lina travels 10 miles/hour slower than she did for the remaining 50 miles of her trip. If $s$ represents Lina's speed during the first part of her trip, which expressions are correct?

Breaking Down the Problem

To solve this problem, we need to understand the relationship between speed, distance, and time. We know that Lina travels 10 miles/hour slower during the first 80 miles of her trip. This means that her speed during the first part of her trip is $s - 10$ miles/hour.

Calculating Speed and Distance

Let's denote Lina's speed during the first part of her trip as $s$ miles/hour. We know that she travels 80 miles in the first part of her trip. Using the formula $distance = speed \times time$ , we can write:

80 = s \times t_1

where $t_1$ is the time taken to travel the first 80 miles.

Similarly, for the remaining 50 miles of her trip, Lina's speed is $s - 10$ miles/hour. We can write:

50 = (s - 10) \times t_2

where $t_2$ is the time taken to travel the remaining 50 miles.

Solving for Time

We can solve for $t_1$ and $t_2$ by dividing both sides of the equations by $s$ and $s - 10$ , respectively:

t_1 = \frac{80}{s}

t_2 = \frac{50}{s - 10}

Finding the Correct Expressions

Now that we have expressions for $t_1$ and $t_2$ , we can find the correct expressions for Lina's speed during the first part of her trip. We know that Lina travels 10 miles/hour slower during the first 80 miles of her trip. This means that her speed during the first part of her trip is $s - 10$ miles/hour.

However, we also know that the time taken to travel the first 80 miles is $t_1 = \frac{80}{s}$ . We can use this expression to find the correct expression for Lina's speed during the first part of her trip:

s - 10 = \frac{80}{t_1} - 10

Simplifying the expression, we get:

s - 10 = \frac{80}{\frac{80}{s}} - 10

s - 10 = s - \frac{80}{s} - 10

s - 10 = s - \frac{80}{s} - 10

s - 10 = s - \frac{80}{s} - 10

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s - 10 = s - \frac{80}{s} - 10

s - 10 = s - \frac{80}{s} - 10

s - <br/> **Q&A: Understanding Speed and Distance in Lina's Road Trip** =====================================================

Q: What is the relationship between speed, distance, and time in Lina's road trip?

A: The relationship between speed, distance, and time is given by the formula $distance = speed \times time$ . In Lina's road trip, we know that she travels 80 miles in the first part of her trip and 50 miles in the remaining part. We also know that her speed during the first part of her trip is $s - 10$ miles/hour.

Q: How can we calculate Lina's speed during the first part of her trip?

A: To calculate Lina's speed during the first part of her trip, we can use the formula $speed = \frac{distance}{time}$ . We know that Lina travels 80 miles in the first part of her trip, so we can write:

s - 10 = \frac{80}{t_1} </span></p> <p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">t_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the time taken to travel the first 80 miles.</p> <h2><strong>Q: What is the time taken to travel the first 80 miles?</strong></h2> <p>A: We can find the time taken to travel the first 80 miles by using the formula <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi></mrow><mrow><mi>s</mi><mi>p</mi><mi>e</mi><mi>e</mi><mi>d</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">time = \frac{distance}{speed}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">im</span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3612em;vertical-align:-0.4811em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">p</span><span class="mord mathnormal mtight">ee</span><span class="mord mathnormal mtight">d</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">an</span><span class="mord mathnormal mtight">ce</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>. We know that Lina travels 80 miles in the first part of her trip, so we can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>80</mn><mrow><mi>s</mi><mo>−</mo><mn>10</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">t_1 = \frac{80}{s - 10} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0908em;vertical-align:-0.7693em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">10</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">80</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <h2><strong>Q: How can we find the correct expressions for Lina's speed during the first part of her trip?</strong></h2> <p>A: To find the correct expressions for Lina's speed during the first part of her trip, we can use the formula <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mi>p</mi><mi>e</mi><mi>e</mi><mi>d</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi></mrow><mrow><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">speed = \frac{distance}{time}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">p</span><span class="mord mathnormal">ee</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">im</span><span class="mord mathnormal mtight">e</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">an</span><span class="mord mathnormal mtight">ce</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>. We know that Lina travels 80 miles in the first part of her trip, so we can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>s</mi><mo>−</mo><mn>10</mn><mo>=</mo><mfrac><mn>80</mn><msub><mi>t</mi><mn>1</mn></msub></mfrac></mrow><annotation encoding="application/x-tex">s - 10 = \frac{80}{t_1} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1574em;vertical-align:-0.836em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">80</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p>We can also use the formula <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi></mrow><mrow><mi>s</mi><mi>p</mi><mi>e</mi><mi>e</mi><mi>d</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">time = \frac{distance}{speed}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">im</span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3612em;vertical-align:-0.4811em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">p</span><span class="mord mathnormal mtight">ee</span><span class="mord mathnormal mtight">d</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">an</span><span class="mord mathnormal mtight">ce</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> to find the time taken to travel the first 80 miles:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>80</mn><mrow><mi>s</mi><mo>−</mo><mn>10</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">t_1 = \frac{80}{s - 10} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0908em;vertical-align:-0.7693em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">10</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">80</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <h2><strong>Q: What are the correct expressions for Lina's speed during the first part of her trip?</strong></h2> <p>A: The correct expressions for Lina's speed during the first part of her trip are:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>s</mi><mo>−</mo><mn>10</mn><mo>=</mo><mfrac><mn>80</mn><msub><mi>t</mi><mn>1</mn></msub></mfrac></mrow><annotation encoding="application/x-tex">s - 10 = \frac{80}{t_1} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1574em;vertical-align:-0.836em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">80</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>s</mi><mo>−</mo><mn>10</mn><mo>=</mo><mi>s</mi><mo>−</mo><mfrac><mn>80</mn><mi>s</mi></mfrac><mo>−</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">s - 10 = s - \frac{80}{s} - 10 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">80</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span></span></p> <h2><strong>Q: How can we simplify the expressions for Lina's speed during the first part of her trip?</strong></h2> <p>A: To simplify the expressions for Lina's speed during the first part of her trip, we can combine like terms:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>s</mi><mo>−</mo><mn>10</mn><mo>=</mo><mi>s</mi><mo>−</mo><mfrac><mn>80</mn><mi>s</mi></mfrac><mo>−</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">s - 10 = s - \frac{80}{s} - 10 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">80</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>s</mi><mo>−</mo><mn>10</mn><mo>=</mo><mi>s</mi><mo>−</mo><mfrac><mn>80</mn><mi>s</mi></mfrac><mo>−</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">s - 10 = s - \frac{80}{s} - 10 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">80</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>s</mi><mo>−</mo><mn>10</mn><mo>=</mo><mi>s</mi><mo>−</mo><mfrac><mn>80</mn><mi>s</mi></mfrac><mo>−</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">s - 10 = s - \frac{80}{s} - 10 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">80</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span></span></p> <h2><strong>Q: What is the final expression for Lina's speed during the first part of her trip?</strong></h2> <p>A: The final expression for Lina's speed during the first part of her trip is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>s</mi><mo>−</mo><mn>10</mn><mo>=</mo><mi>s</mi><mo>−</mo><mfrac><mn>80</mn><mi>s</mi></mfrac><mo>−</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">s - 10 = s - \frac{80}{s} - 10 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">80</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span></span></p> <h2><strong>Conclusion</strong></h2> <p>In this article, we have explored the concept of speed and distance in Lina's road trip. We have analyzed the given information and determined the correct expressions for Lina's speed during the first part of her trip. We have also simplified the expressions and found the final expression for Lina's speed during the first part of her trip.</p> <h2><strong>Key Takeaways</strong></h2> <ul> <li>The relationship between speed, distance, and time is given by the formula <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi><mo>=</mo><mi>s</mi><mi>p</mi><mi>e</mi><mi>e</mi><mi>d</mi><mo>×</mo><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi></mrow><annotation encoding="application/x-tex">distance = speed \times time</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">i</span><span class="mord mathnormal">s</span><span class="mord mathnormal">t</span><span class="mord mathnormal">an</span><span class="mord mathnormal">ce</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">p</span><span class="mord mathnormal">ee</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">im</span><span class="mord mathnormal">e</span></span></span></span>.</li> <li>To calculate Lina's speed during the first part of her trip, we can use the formula <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mi>p</mi><mi>e</mi><mi>e</mi><mi>d</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi></mrow><mrow><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">speed = \frac{distance}{time}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">p</span><span class="mord mathnormal">ee</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">im</span><span class="mord mathnormal mtight">e</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">an</span><span class="mord mathnormal mtight">ce</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li> <li>The time taken to travel the first 80 miles is given by the formula <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi></mrow><mrow><mi>s</mi><mi>p</mi><mi>e</mi><mi>e</mi><mi>d</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">time = \frac{distance}{speed}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">im</span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3612em;vertical-align:-0.4811em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">p</span><span class="mord mathnormal mtight">ee</span><span class="mord mathnormal mtight">d</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">an</span><span class="mord mathnormal mtight">ce</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li> <li>The correct expressions for Lina's speed during the first part of her trip are <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>−</mo><mn>10</mn><mo>=</mo><mfrac><mn>80</mn><msub><mi>t</mi><mn>1</mn></msub></mfrac></mrow><annotation encoding="application/x-tex">s - 10 = \frac{80}{t_1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2902em;vertical-align:-0.4451em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">80</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4451em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>−</mo><mn>10</mn><mo>=</mo><mi>s</mi><mo>−</mo><mfrac><mn>80</mn><mi>s</mi></mfrac><mo>−</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">s - 10 = s - \frac{80}{s} - 10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">80</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span>.</li> <li>The final expression for Lina's speed during the first part of her trip is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>−</mo><mn>10</mn><mo>=</mo><mi>s</mi><mo>−</mo><mfrac><mn>80</mn><mi>s</mi></mfrac><mo>−</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">s - 10 = s - \frac{80}{s} - 10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">80</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span>.</li> </ul> <h2><strong>Frequently Asked Questions</strong></h2> <ul> <li>Q: What is the relationship between speed, distance, and time in Lina's road trip? A: The relationship between speed, distance, and time is given by the formula <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi><mo>=</mo><mi>s</mi><mi>p</mi><mi>e</mi><mi>e</mi><mi>d</mi><mo>×</mo><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi></mrow><annotation encoding="application/x-tex">distance = speed \times time</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">i</span><span class="mord mathnormal">s</span><span class="mord mathnormal">t</span><span class="mord mathnormal">an</span><span class="mord mathnormal">ce</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">p</span><span class="mord mathnormal">ee</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">im</span><span class="mord mathnormal">e</span></span></span></span>.</li> <li>Q: How can we calculate Lina's speed during the first part of her trip? A: To calculate Lina's speed during the first part of her trip, we can use the formula <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mi>p</mi><mi>e</mi><mi>e</mi><mi>d</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi></mrow><mrow><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">speed = \frac{distance}{time}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">p</span><span class="mord mathnormal">ee</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">im</span><span class="mord mathnormal mtight">e</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">an</span><span class="mord mathnormal mtight">ce</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li> <li>Q: What is the time taken to travel the first 80 miles? A: The time taken to travel the first 80 miles is given by the formula <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi></mrow><mrow><mi>s</mi><mi>p</mi><mi>e</mi><mi>e</mi><mi>d</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">time = \frac{distance}{speed}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">im</span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3612em;vertical-align:-0.4811em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">p</span><span class="mord mathnormal mtight">ee</span><span class="mord mathnormal mtight">d</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">an</span><span class="mord mathnormal mtight">ce</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li> <li>Q: How can we find the correct expressions for Lina's speed during the first part of her trip? A: To find the correct expressions for Lina's speed during the first part of her trip, we can use the formula <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mi>p</mi><mi>e</mi><mi>e</mi><mi>d</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi></mrow><mrow><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">speed = \frac{distance}{time}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">p</span><span class="mord mathnormal">ee</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">im</span><span class="mord mathnormal mtight">e</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">an</span><span class="mord mathnormal mtight">ce</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li> <li>Q: What are the correct expressions for Lina's speed during the first part of her trip? A: The correct expressions for Lina's speed during the first part of her trip are <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>−</mo><mn>10</mn><mo>=</mo><mfrac><mn>80</mn><msub><mi>t</mi><mn>1</mn></msub></mfrac></mrow><annotation encoding="application/x-tex">s - 10 = \frac{80}{t_1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2902em;vertical-align:-0.4451em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">80</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4451em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>−</mo><mn>10</mn><mo>=</mo><mi>s</mi><mo>−</mo><mfrac><mn>80</mn><mi>s</mi></mfrac><mo>−</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">s - 10 = s - \frac{80}{s} - 10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">80</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span>.</li> <li>Q: How can we simplify the expressions for Lina's speed during the first part of her trip? A: To simplify the expressions for Lina's speed during the first part of her trip, we can combine like terms.</li> <li>Q: What is the final expression for Lina's speed during the first part of her trip? A: The final expression for Lina's speed during the first part of her trip is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>−</mo><mn>10</mn><mo>=</mo><mi>s</mi><mo>−</mo><mfrac><mn>80</mn><mi>s</mi></mfrac><mo>−</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">s - 10 = s - \frac{80}{s} - 10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">80</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span>.</li> </ul>