The Circumference Of A Circle Is Increasing At The Rate Of 0.5 Meters/minute. What Is The Rate Of Change Of The Area Of The Circle When The Radius Is 4 Meters?A. $2 \, M^2/\text{min}$B. $4 \, M^2/\text{min}$C. $4\pi \,

The Circumference of a Circle: A Mathematical Analysis

In mathematics, the study of rates of change is a fundamental concept that has numerous applications in various fields, including physics, engineering, and economics. One of the most important rates of change is the rate of change of the area of a circle, which is a critical concept in geometry and calculus. In this article, we will explore the rate of change of the area of a circle when its circumference is increasing at a given rate.

The Formula for the Circumference of a Circle

The circumference of a circle is given by the formula:

C = 2πr

where C is the circumference, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.

The Rate of Change of the Circumference

We are given that the circumference of the circle is increasing at a rate of 0.5 meters/minute. This means that the derivative of the circumference with respect to time (t) is:

dC/dt = 0.5 m/min

The Formula for the Area of a Circle

The area of a circle is given by the formula:

A = πr^2

where A is the area, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.

The Rate of Change of the Area

To find the rate of change of the area of the circle, we need to find the derivative of the area with respect to time (t). Using the chain rule, we can write:

dA/dt = d(πr^2)/dt = 2πr(dr/dt)

Given Information

We are given that the radius of the circle is 4 meters. We can substitute this value into the formula for the rate of change of the area:

dA/dt = 2π(4)(dr/dt) = 8π(dr/dt)

Finding the Rate of Change of the Radius

We are given that the circumference of the circle is increasing at a rate of 0.5 meters/minute. We can use this information to find the rate of change of the radius. Using the formula for the circumference, we can write:

dC/dt = d(2πr)/dt = 2π(dr/dt)

Substituting the given value for the rate of change of the circumference, we get:

0.5 = 2π(dr/dt)

Solving for dr/dt, we get:

dr/dt = 0.5 / (2π) = 0.0796 m/min

Finding the Rate of Change of the Area

Now that we have found the rate of change of the radius, we can substitute this value into the formula for the rate of change of the area:

dA/dt = 8π(0.0796) = 2.01 m^2/min

In conclusion, we have found that the rate of change of the area of the circle is approximately 2.01 m^2/min when the radius is 4 meters and the circumference is increasing at a rate of 0.5 meters/minute.

The correct answer is A. $2 \, m^2/\text{min}$
The Circumference of a Circle: A Mathematical Analysis - Q&A

In our previous article, we explored the rate of change of the area of a circle when its circumference is increasing at a given rate. We found that the rate of change of the area is approximately 2.01 m^2/min when the radius is 4 meters and the circumference is increasing at a rate of 0.5 meters/minute. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the formula for the circumference of a circle?

A: The formula for the circumference of a circle is:

C = 2πr

where C is the circumference, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.

Q: What is the rate of change of the circumference of a circle?

A: The rate of change of the circumference of a circle is given by the derivative of the circumference with respect to time (t):

dC/dt = 0.5 m/min

Q: What is the formula for the area of a circle?

A: The formula for the area of a circle is:

A = πr^2

where A is the area, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.

Q: How do I find the rate of change of the area of a circle?

A: To find the rate of change of the area of a circle, you need to find the derivative of the area with respect to time (t). Using the chain rule, you can write:

dA/dt = d(πr^2)/dt = 2πr(dr/dt)

Q: What is the rate of change of the radius of a circle?

A: The rate of change of the radius of a circle is given by the derivative of the radius with respect to time (t):

dr/dt = 0.0796 m/min

Q: How do I find the rate of change of the area of a circle when the radius is 4 meters and the circumference is increasing at a rate of 0.5 meters/minute?

A: To find the rate of change of the area of a circle when the radius is 4 meters and the circumference is increasing at a rate of 0.5 meters/minute, you need to substitute the given values into the formula for the rate of change of the area:

dA/dt = 8π(0.0796) = 2.01 m^2/min

Q: What is the correct answer to the problem?

A: The correct answer is A. $2 \, m^2/\text{min}$

In conclusion, we have answered some frequently asked questions related to the topic of the rate of change of the area of a circle. We hope that this article has been helpful in clarifying any doubts you may have had.

If you are interested in learning more about the topic of the rate of change of the area of a circle, we recommend the following resources:

• Calculus textbooks
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• Math websites and forums

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