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