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Chapter 2: Problem 76

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 \( \pi (6^{2}) - \pi (5^{2}) = 11\pi \)

Step by step solution

01

- Understand the Equations

The given equations are \((x-2)^{2}+(y+3)^{2}=25\) and \((x-2)^{2}+(y+3)^{2}=36\). Each equation represents a circle, and the numbers 25 and 36 are the radii squared of their respective circles.
02

- Find the Radii of the Circles

Take the square root of the radii squared to get the actual radii. The radius of the smaller circle is \( \sqrt{25} = 5 \) and the radius of the larger circle is \( \sqrt{36} = 6 \).
03

- Find the Difference of Areas

Calculate the areas of both circles using the formula \( \pi r^{2} \). Afterwards, subtract the area of the smaller circle from the area of the larger circle to get the area of the donut-shaped region: \( \pi (6^{2}) - \pi (5^{2}) \).
04

- Simplify the result

Calculate the difference from the previous step, this will give the area of the donut-shaped region.

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