A rectangle is inscribed in a circle of radius 5 cm. If the area of the rectangle is equal to 1/3 the area of the circle, what is the perimeter of the rectangle? Express your answer as a decimal to the nearest tenth.
+129899 Call the sides of the rectangle a, b
The diagonal of the rectangle = 2r = 10
And
a^2 + b^2 = 100
The area of the circle = 25pi
So.....the area of the rectangle = (25/3) pi = ab → b = (25pi) / (3a) ...b^2 = 625 pi^2 / (9a^2)
So
a^2 + 625pi^2 / (9a^2) = 100 multiply through by a^2 and rearrange
a^4 - 100a^2 + (625/9)pi^2 = 0
a^4 -100a^2 = - (625pi^2)/9 complete the square on a
a^4 -100a^2 + 2500 = -(625/9) pi^2 + 2500
(a^2 - 50)^2 = - (625/9)pi^2 + 2500 take the positive root
a^2 - 50 = sqrt [- (625/9)pi^2 + 2500 ] ≈ 42.6 cm
a^2 = 42.6 + 50
a^2 ≈ 92.6
a = sqrt (92.6) ≈ 9.6 cm
b = sqrt ( 100 - 92.6) ≈ 2.7 cm
Perimeter of rectangle = 2 ( a + b) = 2 ( 9.6 + 2.7) ≈ 24.6 cm
