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

So the question wants to calculate the change of the area among the circles in respect to its radius changes and the best answer would be the following:
A. 50.27 to 78.54
B.50.27 to 63.62
C.50.27 to 52. 81

The rate of change are the following:
A. 56.24%
B. 26.56%
C. 5.05%

Answer 2

Answer:

a)1. 28.26

2. 26.70

3. 25.44

b) 25.13

Step-by-step explanation:

We are given the following information in the question:

Area of circle

$A(r) = \pi r^2$, where r is the radius of circle.

a) Formula:

Rate of change of area of circle = $\displaystyle\frac{A(r_2) - A(r_1)}{r_2-r_1}$

Putting the given values we get:

1. Rate of change of circle when radius changes from 4 to 5

$\displaystyle\frac{\pi (5)^2- \pi (4)^2}{5-4} = 9\pi = 28.26$

2. Rate of change of circle when radius changes from 4 to 4.5

$\displaystyle\frac{\pi (4.5)^2- \pi (4)^2}{4.5-4} =26.70$

3. Rate of change of circle when radius changes from 4 to 4.1

$\displaystyle\frac{\pi (4.1)^2- \pi (4)^2}{4.1-4} =25.44$

b) Instantaneous rate of change

$\displaystyle\frac{d(A(r))}{dr} = \frac{d(\pi r^2)}{dr} = 2\pi r$

When r = 4

$A'(4) = 2\times \pi \times 4 = 25.13$