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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 \(11\pi\) square units.

Step by step solution

01

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 \(\sqrt{25}=5\).
02

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 \(\pi * 5^{2}=25\pi\).
03

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. \(36\pi - 25\pi = 11\pi\)

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Most popular questions from this chapter

What does it mean if a function \(f\) is increasing on an interval?

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)--2|x+4|+1 $$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. In addition, graph the line \(y-x\) and visually determine if \(f\) and g are inverses. $$ f(x)=\frac{1}{x}+2, g(x)=\frac{1}{x-2} $$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. In addition, graph the line \(y-x\) and visually determine if \(f\) and g are inverses. $$ f(x)=4 x+4, g(x)=0.25 x-1 $$

Perform the indicated operation or operations. $$ (f(x))^{2}-2 f(x)+6, \text { where } f(x)-3 x-4 $$
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