Finding the Range of Values for x

Oct 13, 2025 by windailys.com 88 views

Introduction

When it comes to finding the volume of a box, we often think of it as a straightforward calculation: length x width x height. However, in real-world scenarios, we may be given certain constraints or requirements, such as a specific volume range. In this article, we will explore how to find the values of x that will give a box with a volume between 300 cm^3 and 900 cm^3, given the dimensions 3 cm x 5 cm x 11 cm.

Understanding the Problem

To start, let's break down the problem and understand what we are trying to achieve. We have a box with dimensions 3 cm x 5 cm x 11 cm, and we want to find the values of x that will give a box with a volume between 300 cm^3 and 900 cm^3. This means that we need to find the range of values for x that will result in a volume within this specified range.

The Formula for Volume

The formula for the volume of a box is:

V = l x w x h

where V is the volume, l is the length, w is the width, and h is the height. In this case, we are given the dimensions 3 cm x 5 cm x 11 cm, so we can plug these values into the formula:

V = 3 x 5 x 11

V = 165 cm^3

However, we want to find the values of x that will give a box with a volume between 300 cm^3 and 900 cm^3. This means that we need to find the range of values for x that will result in a volume within this specified range.

Finding the Range of Values for x

To find the range of values for x, we can start by setting up an inequality based on the volume formula:

300 ≤ 3 x 5 x x ≤ 900

Simplifying the inequality, we get:

300 ≤ 15x ≤ 900

Now, we can divide all three parts of the inequality by 15 to get:

20 ≤ x ≤ 60

This means that the values of x that will give a box with a volume between 300 cm^3 and 900 cm^3 are between 20 and 60.

Interpreting the Results

So, what do these results mean in practical terms? If we want to create a box with a volume between 300 cm^3 and 900 cm^3, we can choose any value of x between 20 and 60. For example, if we choose x = 30, the volume of the box would be:

V = 3 x 5 x 30

V = 450 cm^3

This is within the specified range, so x = 30 is a valid solution.

Conclusion

In this article, we explored how to find the values of x that will give a box with a volume between 300 cm^3 and 900 cm^3, given the dimensions 3 cm x 5 cm x 11 cm. We started by understanding the problem and breaking it down into smaller parts, and then we used the formula for volume to set up an inequality. By solving the inequality, we found the range of values for x that will result in a volume within the specified range. We also interpreted the results in practical terms, showing how to choose a valid value of x to create a box with the desired volume.

Additional Examples

To further illustrate the concept, let's consider a few additional examples:

If we want to create a box with a volume between 100 cm^3 and 500 cm^3, we can use the same formula and inequality to find the range of values for x. In this case, the inequality would be: 100 ≤ 3 x 5 x x ≤ 500 Simplifying the inequality, we get: 6.67 ≤ x ≤ 33.33
If we want to create a box with a volume between 500 cm^3 and 2000 cm^3, we can use the same formula and inequality to find the range of values for x. In this case, the inequality would be: 500 ≤ 3 x 5 x x ≤ 2000 Simplifying the inequality, we get: 33.33 ≤ x ≤ 133.33

Real-World Applications

The concept of finding the range of values for x to create a box with a specific volume has many real-world applications. For example:

In manufacturing, knowing the range of values for x can help designers create boxes with the desired volume and shape.
In logistics, understanding the range of values for x can help companies optimize their packaging and shipping processes.
In engineering, knowing the range of values for x can help designers create structures with the desired volume and strength.

Conclusion

Q: What is the formula for the volume of a box?

A: The formula for the volume of a box is V = l x w x h, where V is the volume, l is the length, w is the width, and h is the height.

Q: How do I find the range of values for x to create a box with a specific volume?

A: To find the range of values for x, you can set up an inequality based on the volume formula. For example, if you want to create a box with a volume between 300 cm^3 and 900 cm^3, you can set up the inequality 300 ≤ 3 x 5 x x ≤ 900. Simplifying the inequality, you get 20 ≤ x ≤ 60.

Q: What if I want to create a box with a volume between 100 cm^3 and 500 cm^3? How do I find the range of values for x?

A: To find the range of values for x, you can set up an inequality based on the volume formula. For example, if you want to create a box with a volume between 100 cm^3 and 500 cm^3, you can set up the inequality 100 ≤ 3 x 5 x x ≤ 500. Simplifying the inequality, you get 6.67 ≤ x ≤ 33.33.

Q: What if I want to create a box with a volume between 500 cm^3 and 2000 cm^3? How do I find the range of values for x?

A: To find the range of values for x, you can set up an inequality based on the volume formula. For example, if you want to create a box with a volume between 500 cm^3 and 2000 cm^3, you can set up the inequality 500 ≤ 3 x 5 x x ≤ 2000. Simplifying the inequality, you get 33.33 ≤ x ≤ 133.33.

Q: How do I use the range of values for x in real-world applications?

A: The range of values for x can be used in various real-world applications, such as:

In manufacturing, knowing the range of values for x can help designers create boxes with the desired volume and shape.
In logistics, understanding the range of values for x can help companies optimize their packaging and shipping processes.
In engineering, knowing the range of values for x can help designers create structures with the desired volume and strength.

Q: What are some common mistakes to avoid when finding the range of values for x?

A: Some common mistakes to avoid when finding the range of values for x include:

Not setting up the inequality correctly
Not simplifying the inequality correctly
Not considering the units of measurement
Not considering the physical constraints of the problem

Q: How can I practice finding the range of values for x?

A: You can practice finding the range of values for x by:

Working through example problems
Creating your own problems and solving them
Using online resources and calculators to check your work
Discussing the concept with a teacher or tutor

Conclusion

In conclusion, finding the range of values for x to create a box with a specific volume is a fundamental concept in mathematics and has many real-world applications. By understanding the formula for volume and setting up an inequality, we can solve for the range of values for x that will result in a volume within the specified range. Whether it's in manufacturing, logistics, or engineering, knowing the range of values for x can help designers and engineers create structures with the desired volume and shape.