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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.

 Feb 8, 2024
 #1
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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)

 

cool cool cool

 Feb 8, 2024
edited by CPhill  Feb 8, 2024

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