A rectangle whose area is 32 and perimeter is 24 is inscribed in a circle as shown. Find the area of the shaded region.

Guest Jul 2, 2020

#1**0 **

Let L represent the length of the rectangle and W represent the width.

L · W = 32 2L + 2W = 24 ---> L + W = 12 ---> W = 12 - L

L · W = 32 ---> L · (12 - L) = 32 ---> 12L - L^{2} = 32

---> 0 = L^{2} - 12L + 32 ---> 0 = (L - 8)(L - 4)

Length = 8 and Width = 4.

The distance from the center of the circle to the top side = 2.

The distance from the center of the circle to the right side = 4.

The distance from the center of the circle to the top-right corner = r

Using the Pythagorean Theorem: r^{2} = 2^{2} + 4^{2} = 4 + 16 = 20 ---> r = sqrt(20)

The area of the circle is: pi · sqrt(20)^{2} = 20pi

The area of the rectangle is 8 · 4 = 32

The area of the shaded region = 20pi - 32

geno3141 Jul 2, 2020