Question

The rate of change of area of a circle with respect to its circumference when radius is 4cm, is

• A. \( \frac{2 \, \text{cm}^2}{\text{cm}} \)
• B. \( \frac{2 \, \text{cm}^2}{\text{cm}} \)
• C. \( \frac{8 \, \text{cm}^2}{\text{cm}} \)
• D. \( \frac{16 \, \text{cm}^2}{\text{cm}} \)

Hint:

Alternative Method for Differentiating Area with Respect to Circumference:

From the formula for the circumference of a circle, \( C = 2\pi r \), we can express the radius \( r \) as a function of \( C \):

\[ r = \frac{C}{2\pi} \] Substitute this into the area formula \( A = \pi r^2 \) to get the area as a function of \( C \): \[ A = \pi \left( \frac{C}{2\pi} \right)^2 = \pi \left( \frac{C^2}{4\pi^2} \right) = \frac{C^2}{4\pi} \] Now, differentiate \( A \) with respect to \( C \): \[ \frac{dA}{dC} = \frac{2C}{4\pi} = \frac{C}{2\pi} \] Substitute \( C = 2\pi r = 2\pi(4) = 8\pi \): \[ \frac{dA}{dC} = \frac{8\pi}{2\pi} = 4 \]

Final Answer: The value of \( \frac{dA}{dC} \) is 4.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:

We are asked to find the derivative of the area \( A \) with respect to the circumference \( C \), which is represented as \( \frac{dA}{dC} \). This is a related rates problem where both \( A \) and \( C \) are functions of the radius \( r \).

Step 2: Key Formula or Approach:

Let \( A \) be the area and \( C \) be the circumference of a circle with radius \( r \). 
\[ A = \pi r^2 \] \[ C = 2\pi r \] We can use the chain rule to find \( \frac{dA}{dC} \): \[ \frac{dA}{dC} = \frac{dA/dr}{dC/dr} \]

Step 3: Detailed Explanation:

First, we find the derivatives of \( A \) and \( C \) with respect to \( r \):

\[ \frac{dA}{dr} = \frac{d}{dr}(\pi r^2) = 2\pi r \] \[ \frac{dC}{dr} = \frac{d}{dr}(2\pi r) = 2\pi \] Now, we can find \( \frac{dA}{dC} \) using the chain rule: \[ \frac{dA}{dC} = \frac{2\pi r}{2\pi} = r \] The problem asks for this rate of change when the radius is 4 cm. 
Substituting \( r = 4 \) cm: \[ \frac{dA}{dC} = 4 \text{ cm} \] The units of this rate are area units divided by length units, which is cm\(^2\)/cm. So the result should just be cm. However, the options provided all have units cm\(^2\)/cm, which is a bit redundant but guides us to the numerical answer.
The final numerical value is 4.

Step 4: Final Answer:

The rate of change of the area with respect to its circumference is 4 cm²/cm.