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.
Step 1: Problem Definition:
The objective is to determine the derivative of the area \( A \) with respect to the circumference \( C \), denoted as \( \frac{dA}{dC} \). This is a related rates problem where both \( A \) and \( C \) are functions of the radius \( r \).
Step 2: Relevant Formulas:
For a circle with radius \( r \), the area \( A \) and circumference \( C \) are given by:
\[ A = \pi r^2 \] \[ C = 2\pi r \] The derivative \( \frac{dA}{dC} \) can be found using the chain rule:
\[ \frac{dA}{dC} = \frac{dA/dr}{dC/dr} \]
Step 3: Calculation and Evaluation:
First, calculate the derivatives of \( A \) and \( C \) with respect to \( r \):
\[ \frac{dA}{dr} = \frac{d}{dr}(\pi r^2) = 2\pi r \] \[ \frac{dC}{dr} = \frac{d}{dr}(2\pi r) = 2\pi \] Apply the chain rule:
\[ \frac{dA}{dC} = \frac{2\pi r}{2\pi} = r \]
The problem specifies that the radius is 4 cm. Substitute \( r = 4 \) cm:
\[ \frac{dA}{dC} = 4 \text{ cm} \]
The units are cm\(^2\)/cm. The numerical value is 4.
Step 4: Final Result:
The rate of change of the area with respect to its circumference is 4 cm2/cm.