Question

there are 8 problems in this section, but you only need to complete 5. if you do more, you will receive extra points for the assignment. let a be the area of an expanding circle with radius r. a = πr² if the radius increases at a rate given by \\( \frac{dr}{dt}=3 \\) cm per second, find the rate at which the area is increasing, in sq.centimeters per second, when r = 1. \\( \frac{da}{dt}= \\) question help: video message instructor submit question jump to answer

Explanation:

Step1: Differentiate the area formula with respect to \(t\)

Using the chain - rule, if \(A = \pi r^{2}\), then \(\frac{dA}{dt}=2\pi r\frac{dr}{dt}\)

Step2: Substitute the given values

We know that \(\frac{dr}{dt} = 3\) cm/s and \(r = 1\) cm.
Substitute these values into the formula \(\frac{dA}{dt}=2\pi r\frac{dr}{dt}\)
\(\frac{dA}{dt}=2\pi\times1\times3\)

Answer:

\(6\pi\)