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
(type an exact answer, using π as needed.)

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 \(dA=A^{\prime}(r)dr\). Differentiating \(A(r)=\pi r^{2}\) with respect to \(r\), we get \(A^{\prime}(r) = 2\pi r\).

Step2: Identify the values of \(r\) and \(dr\)

We are given that \(r = 10.00\) m and \(dr=\Delta r=10.09 - 10.00=0.09\) m.

Step3: Calculate the differential \(dA\)

Substitute \(r = 10\) and \(dr = 0.09\) into the formula \(dA=2\pi rdr\).

$$
LATEXBLOCK0
$$

Answer:

\(1.8\pi\)