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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.

 Jul 15, 2022
 #1
avatar+124525 
+1

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

 

cool cool cool

 Jul 15, 2022
 #2
avatar+1105 
+7

Here's my go at this!

 

Area of circle: \(\pi r^2\)= 6.25pi

Rectangle area = 1/3 of 6.25pi, right? So, here's an equation. Let's call the height of the rectangle x.

 

\(5x=6.25\pi\)

Just solve for x!

 Jul 15, 2022

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