+556 A rectangle is inscribed in a circle of radius $5$ cm. If the area of the rectangle is equal to 24, what is the perimeter of the rectangle? Express your answer as a decimal to the nearest tenth.
+129830 The length of the diagonal of the rectangle = the diameter of the circle = 10
So....using the P Theorem
L^2 + W^2 = 10^2 (1)
Area = LW = 24
W = 24/L (2)
Sub (2) into (1)
L^2 + (24/L)^2 =10^2
L^2 + 576/L^2 = 100 multiply through by L^2
L^2 + 576 = 100L^2
L^4 - 100 L^2 + 576 = 0 complete the square on L
L^4 - 100L^2 + 2500 = -576 + 2500
(L^2 - 50)^2 = 1924 take the positive root (1)
L^2 - 50 = sqrt (1924)
L^2 = sqrt (1924) + 50
L = sqrt [ sqrt (1924) + 50 ] ≈ 9.688
W = 24/ 9.688 ≈ 2.477
Perimeter = 2 (W + L) = 2 (9.688 + 2.477) = 24.3
BTW - we would get the same result by taking the negative root in (1)
