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# Circles

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Circle A is in the interior of circle B. The diameter of circle B is 16 cm. What is the diameter of circle A for which the ratio of the shaded area to the area of circle A is 8:1? Apr 20, 2021

#1
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Note  that  we  could move A  to  B  and have concentric circles

The radius  of  the larger circle =  8

We  are looking for this

Area of  circle B  -  area  of  circle A   =  shaded area

And

Shaded  area  =  8 * area  of circle A....so

Area of  circle B  -  area of circle A  =  8 *  area ofcircle A

Area of  circle B  =  9 *  area of circle A

pi * 8^2  = 9 *pi * r^2          where  r  is  the radius  of circle A

64  = 9 r^2

64/9  = r^2                 take  the positive root of both sides

8 / 3   cm   = r  =  radius  of smaller circle A

So....the diameter of  circle A = 2 (8/3)   =  16 / 3   cm   Apr 20, 2021
edited by CPhill  Apr 20, 2021
#2
+1

Circle B area = 64pi

Circle A area = (64pi) / 9

Circle A radius = sqrt(64 / 9)

Circle A diameter = 2[sqrt(64 / 9)]

D = 5 1/3 cm

Apr 21, 2021
edited by civonamzuk  Apr 21, 2021