Question

Expanding Circle The radius of a circle is increased from 2.00 to 2.02 \(\mathrm{m} .\) (a) Estimate the resulting change in area. (b) Estimate as a percentage of the circle's original area.

Short answer

The area of the circle increased by approximately \(0.0804π square meters\) or about 2.01%.

Step-by-step solution

Calculate original area

First, let's calculate the original area of the circle with a radius of 2.00 m using the formula \(A = πr^2\). The original area \(A_1\) then becomes \(A_1 = π(2.00^2) = 4π square meters.\)

Calculate new area

Next, let's calculate the new area of the circle with a radius of 2.02 m using the formula \(A = πr^2\). The new area \(A_2\) then becomes \(A_2 = π(2.02^2) = 4.0804π square meters.\)

Calculate the change in area

The change in area (\(ΔA\)) is new area minus original area, that is, \(ΔA = A_2 - A_1 = 4.0804π - 4π = 0.0804π square meters.\)

Calculate the percentage change

The percentage change in area can be calculated using the formula \((\Delta A / A_1) * 100\). Therefore, the percentage change is \((0.0804π / 4π) * 100 = 2.01%.\)