Solving for Constant of Proportionality

Oct 13, 2025 by windailys.com 143 views

Problem: Understanding Direct and Inverse Proportionality Relationships

In mathematics, proportionality relationships are used to describe the relationship between two or more variables. Direct proportionality occurs when one variable increases as the other variable increases, while inverse proportionality occurs when one variable decreases as the other variable increases. In this article, we will explore two given relationships and use them to solve for unknown variables.

The first relationship states that ${$ p $}$ is directly proportional to the square root of ${$ w $}$. This can be expressed mathematically as:

${$ p = k \sqrt{w} $}$

where ${$ k $}$ is the constant of proportionality.

Given Information

We are given that ${$ p = 42 $}$ when ${$ w = 36 $}$. We can use this information to solve for the constant of proportionality, ${$ k $}$.

Solving for Constant of Proportionality

Substituting the given values into the equation, we get:

${$ 42 = k \sqrt{36} $}$

Simplifying the equation, we get:

${$ 42 = k \cdot 6 $}$

Dividing both sides by 6, we get:

${$ k = 7 $}$

The second relationship states that ${$ w $}$ is inversely proportional to the cube of ${$ t $}$. This can be expressed mathematically as:

${$ w = \frac{k}{t^3} $}$

where ${$ k $}$ is the constant of proportionality.

Given Information

We are given that ${$ w = 24 $}$ when ${$ t = 2 $}$. We can use this information to solve for the constant of proportionality, ${$ k $}$.

Solving for Constant of Proportionality

Substituting the given values into the equation, we get:

${$ 24 = \frac{k}{2^3} $}$

Simplifying the equation, we get:

${$ 24 = \frac{k}{8} $}$

Multiplying both sides by 8, we get:

${$ k = 192 $}$

Now that we have solved for the constants of proportionality in both relationships, we can combine them to solve for the unknown variables.

Solving for ${$ p $}$

We can substitute the value of ${$ k $}$ from the direct proportionality relationship into the equation for ${$ p $}$:

${$ p = 7 \sqrt{w} $}$

Solving for ${$ w $}$

We can substitute the value of ${$ k $}$ from the inverse proportionality relationship into the equation for ${$ w $}$:

${$ w = \frac{192}{t^3} $}$

Solving for ${$ t $}$

We can substitute the value of ${$ w $}$ into the equation for ${$ p $}$:

${$ p = 7 \sqrt{\frac{192}{t^3}} $}$

Simplifying the equation, we get:

${$ p = \frac{7 \sqrt{192}}{t^{\frac{3}{2}}} $}$

Solving for ${$ t $}$

We can substitute the value of ${$ p $}$ into the equation:

${$ 42 = \frac{7 \sqrt{192}}{t^{\frac{3}{2}}} $}$

Simplifying the equation, we get:

${$ t^{\frac{3}{2}} = \frac{7 \sqrt{192}}{42} $}$

Simplifying further, we get:

${$ t^{\frac{3}{2}} = \frac{7 \sqrt{3} \sqrt{64}}{42} $}$

Simplifying further, we get:

${$ t^{\frac{3}{2}} = \frac{7 \cdot 8 \sqrt{3}}{42} $}$

Simplifying further, we get:

${$ t^{\frac{3}{2}} = \frac{56 \sqrt{3}}{42} $}$

Simplifying further, we get:

${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

Simplifying further, we get:

${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

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${$ t^{\frac{3}{2}} = \frac{8 \sqrt{3}}{3} $}$

Simplifying further, we get:

${$ t^{\frac{3}{2}} = \frac{
Problem: Understanding Direct and Inverse Proportionality Relationships - Q&A

In our previous article, we explored two given relationships and used them to solve for unknown variables. In this article, we will answer some frequently asked questions related to direct and inverse proportionality relationships.

Q: What is direct proportionality?

A: Direct proportionality occurs when one variable increases as the other variable increases. This can be expressed mathematically as:

[$ y = kx $}$

where ${$ k $}$ is the constant of proportionality.

Q: What is inverse proportionality?

A: Inverse proportionality occurs when one variable decreases as the other variable increases. This can be expressed mathematically as:

${$ y = \frac{k}{x} $}$

where ${$ k $}$ is the constant of proportionality.

Q: How do I determine if a relationship is direct or inverse proportionality?

A: To determine if a relationship is direct or inverse proportionality, you can use the following steps:

Graph the relationship on a coordinate plane.
If the graph is a straight line with a positive slope, the relationship is direct proportionality.
If the graph is a hyperbola, the relationship is inverse proportionality.

Q: How do I solve for the constant of proportionality?

A: To solve for the constant of proportionality, you can use the following steps:

Write the equation of the relationship.
Substitute the given values into the equation.
Solve for the constant of proportionality.

Q: What is the difference between direct and inverse proportionality?

A: The main difference between direct and inverse proportionality is the relationship between the variables. In direct proportionality, one variable increases as the other variable increases. In inverse proportionality, one variable decreases as the other variable increases.

Q: Can I have multiple constants of proportionality in a relationship?

A: No, you cannot have multiple constants of proportionality in a relationship. The constant of proportionality is a unique value that describes the relationship between the variables.

Q: Can I have a relationship that is both direct and inverse proportionality?

A: No, a relationship cannot be both direct and inverse proportionality at the same time. A relationship is either direct or inverse proportionality, but not both.

Q: How do I apply direct and inverse proportionality in real-life situations?

A: Direct and inverse proportionality are used in many real-life situations, such as:

Physics: To describe the relationship between force and distance.
Engineering: To design systems that require proportional relationships.
Economics: To model the relationship between supply and demand.

In this article, we answered some frequently asked questions related to direct and inverse proportionality relationships. We hope that this article has provided you with a better understanding of these important mathematical concepts.

Khan Academy: Direct and Inverse Proportionality
Mathway: Direct and Inverse Proportionality
Wolfram Alpha: Direct and Inverse Proportionality

Direct and inverse proportionality are fundamental concepts in mathematics that have many real-life applications. By understanding these concepts, you can better analyze and solve problems in various fields. We hope that this article has provided you with a solid foundation in direct and inverse proportionality relationships.