+0

# geometry question

0
127
2

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
+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   Jul 15, 2022
#2
+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