Question

the radius of a circle is increased from 10.00 to 10.09 m.
a. estimate the resulting change in area.
b. express the estimate as a percentage of the circles original area.
a. the estimated change in area is 1.8π m².
(type an exact answer, using π as needed.)
b. the estimated change in area is □% of the circles original area.
(type an integer or a decimal. do not round.)

Explanation:

Step1: Recall the formula for the area of a circle

The area of a circle is \(A = \pi r^{2}\). The differential \(dA\) is given by \(dA=2\pi rdr\). Here, \(r = 10\) (original radius) and \(dr=0.09\) (change in radius).

Step2: Calculate the original area

The original area \(A_{0}=\pi r^{2}=\pi\times(10)^{2}=100\pi\).

Step3: Calculate the percentage change

We know from part (a) that \(dA = 1.8\pi\). The percentage change is \(\frac{dA}{A_{0}}\times100\). Substitute \(dA = 1.8\pi\) and \(A_{0}=100\pi\) into the formula: \(\frac{1.8\pi}{100\pi}\times 100\).

Answer:

\(1.8\)