Not using the correct equation

Oct 13, 2025 by miodeya.com 18 views

Direct Proportionality: Solving for M and r

In mathematics, direct proportionality is a fundamental concept that describes the relationship between two variables. When two variables are directly proportional, it means that as one variable increases or decreases, the other variable also increases or decreases at a constant rate. In this article, we will explore the concept of direct proportionality and use it to solve for the value of M and r in a given problem.

Direct proportionality can be represented mathematically using the equation:

M ∝ r^3

This equation states that M is directly proportional to the cube of r. In other words, as r increases or decreases, M also increases or decreases at a rate that is proportional to the cube of r.

We are given that when r = 2, M = 52. We need to use this information to solve for the value of M when r = 8, and also solve for the value of r when M = 416.

Solving for M when r = 8

To solve for M when r = 8, we can use the given equation:

M ∝ r^3

We can rewrite this equation as:

M = kr^3

where k is a constant of proportionality. We can find the value of k by substituting the given values of r and M:

52 = k(2)^3

Simplifying this equation, we get:

52 = k(8)

Dividing both sides by 8, we get:

k = 6.5

Now that we have found the value of k, we can substitute r = 8 into the equation:

M = 6.5(8)^3

Simplifying this equation, we get:

M = 6.5(512)

M = 3328

Therefore, the value of M when r = 8 is 3328.

Solving for r when M = 416

To solve for r when M = 416, we can use the same equation:

M = kr^3

We can substitute M = 416 and k = 6.5 (which we found earlier) into this equation:

416 = 6.5(r)^3

Dividing both sides by 6.5, we get:

(r)^3 = 64

Taking the cube root of both sides, we get:

r = 4

Therefore, the value of r when M = 416 is 4.

In this article, we used the concept of direct proportionality to solve for the value of M and r in a given problem. We found that when r = 8, M = 3328, and when M = 416, r = 4. This demonstrates the power of direct proportionality in solving real-world problems.

Algebra
Geometry
Trigonometry
Direct Proportionality: Q&A

In our previous article, we explored the concept of direct proportionality and used it to solve for the value of M and r in a given problem. In this article, we will answer some frequently asked questions about direct proportionality and provide additional examples to help you understand this important mathematical concept.

Q: What is direct proportionality?

A: Direct proportionality is a mathematical concept that describes the relationship between two variables. When two variables are directly proportional, it means that as one variable increases or decreases, the other variable also increases or decreases at a constant rate.

Q: How is direct proportionality represented mathematically?

A: Direct proportionality can be represented mathematically using the equation:

M ∝ r^3

This equation states that M is directly proportional to the cube of r.

Q: What is the constant of proportionality?

A: The constant of proportionality is a value that is used to describe the relationship between the two variables. In the equation M = kr^3, k is the constant of proportionality.

Q: How do I find the constant of proportionality?

A: To find the constant of proportionality, you can use the given values of r and M to solve for k. For example, if M = 52 and r = 2, you can substitute these values into the equation:

52 = k(2)^3

Simplifying this equation, you get:

52 = k(8)

Dividing both sides by 8, you get:

k = 6.5

Q: What are some real-world examples of direct proportionality?

A: Direct proportionality is used in many real-world applications, including:

Physics: The force of gravity is directly proportional to the mass of an object.
Engineering: The stress on a material is directly proportional to the force applied to it.
Economics: The demand for a product is directly proportional to the price of the product.

Q: How do I use direct proportionality to solve problems?

A: To use direct proportionality to solve problems, you can follow these steps:

Write the equation that describes the relationship between the two variables.
Substitute the given values of r and M into the equation.
Solve for the constant of proportionality.
Use the constant of proportionality to find the value of M or r.

Q: What are some common mistakes to avoid when working with direct proportionality?

A: Some common mistakes to avoid when working with direct proportionality include:

Not using the correct equation: Make sure to use the correct equation that describes the relationship between the two variables.
Not substituting the correct values: Make sure to substitute the correct values of r and M into the equation.
Not solving for the constant of proportionality: Make sure to solve for the constant of proportionality before using it to find the value of M or r.

In this article, we answered some frequently asked questions about direct proportionality and provided additional examples to help you understand this important mathematical concept. We hope that this article has been helpful in clarifying any confusion you may have had about direct proportionality.