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:
Calculate the derivative of the area \( A \) with respect to the circumference \( C \), denoted as \( \frac{dA}{dC} \). This is a related rates problem where \( A \) and \( C \) are both functions of the radius \( r \).
Step 2: Required Formulas:
For a circle with radius \( r \):
Area \( A = \pi r^2 \)
Circumference \( C = 2\pi r \)
Use the chain rule for differentiation: \( \frac{dA}{dC} = \frac{dA/dr}{dC/dr} \).
Step 3: Calculation:
Differentiate \( A \) 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:
\( \frac{dA}{dC} = \frac{2\pi r}{2\pi} = r \)
Evaluate at \( r = 4 \) cm:
\( \frac{dA}{dC} = 4 \) cm
The units are cm\(^2\)/cm. The numerical value is 4.
Step 4: Conclusion:
The rate of change of the area with respect to its circumference at a radius of 4 cm is 4 cm2/cm.