A rectangle whose area is 32 and perimeter is 24 is inscribed in a circle as shown. Find the area of the shaded region.
+23246 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 - L2 = 32
---> 0 = L2 - 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: r2 = 22 + 42 = 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
