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A rectangle whose area is 32 and perimeter is 24 is inscribed in a circle as shown. Find the area of the shaded region.

 

 Jul 2, 2020
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Let  L  represent the length of the rectangle and  W  represent the width.

 

L · W  =  32          2L + 2W  =  24   --->   L + W  =  12   --->   W  =  12 - L

 

L · W  =  32     --->     L · (12 - L)  =  32     --->     12L - L2  =  32

                       --->     0  =  L2 - 12L + 32     --->     0  =  (L - 8)(L - 4)     

 

Length  =  8     and     Width  =  4.

 

The distance from the center of the circle to the top side = 2.

The distance from the center of the circle to the right side = 4.

The distance from the center of the circle to the top-right corner  =  r

 

Using the Pythagorean Theorem:  r2  =  22 + 42  =  4 + 16  =  20    --->     r  =  sqrt(20)

 

The area of the circle is:  pi · sqrt(20)2  =  20pi

 

The area of the rectangle is  8 · 4  =  32

 

The area of the shaded region  =  20pi - 32

 Jul 2, 2020

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