The radius, r cm, of a circle at time t seconds is given by r = 2t 2 + 1. Express its

area, A cm2 , in terms of t and find the rate of change of the area at the instant when
t=2. (Leave your answer in terms of p )

you know

A = π r^2
replace r with 2t^2 + 1
and expand

differentiate to find the rate of change
plug in t = 2

To find the area of a circle, you can use the formula A = πr^2, where A represents the area and r represents the radius of the circle.

In this case, the radius of the circle is given by r = 2t^2 + 1. Substituting this into the formula for the area, we get:

A = π(2t^2 + 1)^2
= π(4t^4 + 4t^2 + 1)

Therefore, the area of the circle is A = 4πt^4 + 4πt^2 + π.

To find the rate of change of the area at the instant when t = 2, we need to find the derivative of the area function with respect to t, and then evaluate it at t = 2.

Taking the derivative of A with respect to t, we get:

dA/dt = d/dt [4πt^4 + 4πt^2 + π]
= 16πt^3 + 8πt

Now, substituting t = 2 into the derivative expression:

dA/dt (t=2) = 16π(2)^3 + 8π(2)
= 128π + 16π
= 144π

Therefore, the rate of change of the area at the instant when t = 2 is 144π cm^2/s.