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# help with geo

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The radius of the smaller circle in the figure below is 6 units and the area of the overlapping rectangle is 243 sq units.  How many units  is the radius of the larger circle? Jan 5, 2021

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This answer is going to assume that the short side of the rectange is equal to the radius of the large circle.

Our goal is to find the short side of the rectangle. The short side must be between 6 and 12(radius and diameter of small circle). Let's start by finding some factor pairs for 243.

Pair 1: 1*243 (doesn't work)

Pair 2: 3*81(doesn't work)

Pair 3: 9*27(works)

As those are the only three pairs, pair 3 will work.

Now that we know the short side is 9, we can see that the radius of the large circle is 9

Jan 5, 2021
edited by GR0001  Jan 5, 2021
edited by GR0001  Jan 5, 2021
#2
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The distance  between  the  centers of the  circles  = 6  +  R   where R is the radius of  the larger circle

So  we  can  form a right triangle  with   the hypotenuse =   (6 + R)  and  one leg =  R - 6

So.....the  width of the rectangle is    sqrt   [ (6 + R)^2   - ( R - 6)^2 ]   =   sqrt  (24R)

And the height of the rectangle  = R

So

sqrt ( 24R) * R  = 243        square both sides

24R * R ^2   =   243^2

24R^3   = 243^2

R^3  =  243^2 / 24

R  =    (243^2 / 24)^(1/3)  =  27/2  =   13.5   Jan 5, 2021
edited by CPhill  Jan 5, 2021
#3
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The radius of the smaller circle in the figure below is 6 units and the area of the overlapping rectangle is 243 sq units.  How many units is the radius of the larger circle?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

A = 243         r = 6         R = ?

OQ = R + 6           PQ = R - 6           OP = 243 / R

(R + 6)2 - (R - 6)2 = (243 / R)2

R = 13.5 units  Jan 5, 2021