Problem 96 - College Algebra

Question

Find the area of the donut-shaped region bounded by the graphs of \((x-2)^{2}+(y+3)^{2}=25\) and \((x-2)^{2}+(y+3)^{2}=36\).

Step-by-Step Solution

Answer

The area of the donut-shaped region is \(11\pi\) square units.

Step 1: Find the radius of the circles

Get the radius of each circle by taking the square root of the constant term on the right side of each equation. The radius of the outer circle is \(\sqrt{36}=6\) and the radius of inner circle is \(\sqrt{25}=5\).

Step 2: Calculate the area of the circles

Use the formula \(\pi r^{2}\) to find the area of each circle. The area of the outer circle is \(\pi * 6^{2}=36\pi\) and the area of the inner circle is \(\pi * 5^{2}=25\pi\).

Step 3: Calculate the area of the donut-shaped region

Subtract the area of the inner circle from the area of the outer circle to get the area of the donut-shaped region. \(36\pi - 25\pi = 11\pi\)