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Chapter 2: Problem 96

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

Short Answer

Expert verified
The area of the donut-shaped region is \(11\pi\).

Step by step solution

01

Identify the radii of the circles

The circles equations are \((x-2)^{2}+(y+3)^{2}=25\) and \((x-2)^{2}+(y+3)^{2}=36\). So, the radii of these circles are \(\sqrt{25} = 5\) and \(\sqrt{36} = 6\) respectively.
02

Compute the areas of the circles

Using the area formula for circles \(\pi r^{2}\), compute the areas of each circle individually. The area of the first circle is \( \pi (5)^{2} = 25\pi\) and the area of the second circle is \(\pi (6)^{2} = 36\pi\).
03

Subtract the smaller area from the larger

The area of the annulus or donut-shaped region is the difference in the areas of the two circles. This gives \(36\pi - 25\pi = 11\pi\).

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