Question

Rate of change of area of circle per second with respect to its radius \( r \) when \( r = 5 \, \text{cm} \) will be \(\underline{\hspace{2cm}}\)

Hint:

To find the rate of change of area with respect to radius, use the derivative of the area formula and apply the chain rule.

Answer 1

Step 1: Formula for area of a circle.
The area \( A \) of a circle is given by \( A = \pi r^2 \). Differentiating with respect to time \( t \), we get: \[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt}. \]

Step 2: Calculate at \( r = 5 \, \text{cm} \).
Substitute \( r = 5 \, \text{cm} \) into the formula: \[ \frac{dA}{dt} = 2\pi (5) \frac{dr}{dt}. \]

Step 3: Conclusion.
Thus, the rate of change of area is \( 10\pi \frac{dr}{dt} \), where \( \frac{dr}{dt} \) is the rate of change of the radius.