Question

a) Find the average rate of change of the area of a circle with respect to its radius r as r changes from 4 to each of the following.

(i) 4 to 5 _______

(ii) 4 to 4.5 _______

(iii) 4 to 4.1 _______


(b) Find the instantaneous rate of change when r = 4.

A'(4) = ______

Answer 1

Formula for area: \[A = \pi r^2\]
To find average rate of change, find the value of A at both radii and find the difference.

i.) \[A _{1}=\pi (4)^2 = 16\pi\]
\[A _{2}=\pi (5)^2 = 25\pi\]
\[\Delta A = A _2 - A _1 = 25\pi - 16\pi = 9\pi\]

I'll leave ii. and iii. for you to do.

For part b - Find dA/dr and let r = 4.
\[A = \pi r^2\]
\[dA/dr = 2\pi r\]
\[dA/dr (4) = 2\pi (4) = 8\pi\]

Answer 2

Oh, whoops. For the average rate of change, you have to divide the change in area by the change in the radius. The first one is (25pi - 16pi) / (5 - 4) = 9pi / 1 = 9pi. This doesn't change the answer for the first one, but you have to remember that for the other two.

Answer 3

thanks , I got it.