Question

let a be the area of a circle with radius r. if \\(\frac{dr}{dt} = 5\\), find \\(\frac{da}{dt}\\) when \\(r = 5\\).

Explanation:

Step1: Recall the area formula of a circle

The area \( A \) of a circle with radius \( r \) is given by the formula \( A = \pi r^2 \).

Step2: Differentiate both sides with respect to \( t \)

Using the chain rule, we differentiate \( A = \pi r^2 \) with respect to \( t \). The derivative of \( A \) with respect to \( t \) is \( \frac{dA}{dt} \), and for the right - hand side, using the power rule \( \frac{d}{dx}(x^n)=nx^{n - 1} \) and the chain rule, we have \( \frac{d}{dt}(\pi r^2)=2\pi r\frac{dr}{dt} \). So, \( \frac{dA}{dt}=2\pi r\frac{dr}{dt} \).

Step3: Substitute the given values

We are given that \( \frac{dr}{dt} = 5 \) and \( r = 5 \). Substitute these values into the formula for \( \frac{dA}{dt} \):
\( \frac{dA}{dt}=2\pi\times5\times5 \)
\( \frac{dA}{dt}=50\pi \)

Answer:

\( 50\pi \)