Problem 76 - 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 \( \pi (6^{2}) - \pi (5^{2}) = 11\pi \)

Step 1 - 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.

Step 2 - 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 \( \sqrt{36} = 6 \).

Step 3 - 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: \( \pi (6^{2}) - \pi (5^{2}) \).

Step 4 - Simplify the result

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