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The diameter, in inches, of a sphere with twice the volume of a sphere of radius 9 inches can be expressed in the form $$a\sqrt[3]{b}$$ where $$a$$ and $$b$$ are positive integers and $$b$$ contains no perfect cube factors. compute $$a+b$$.

Apr 18, 2019

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Volume of a sphere with radius of  9  = (4/3)pi(9)^3  = 972pi

Volume of our sphere  =  1944 pi

So

1944 pi  = (4/3) pi*r^3

1944 = (4/3) ^ r^3

1458  = r^3

2*3^6  = r^3

Take the cube root of both sides

3^2∛2  =  9∛2  = r

So.....the diameter is twice this  =  18∛2

And a + b  =  20

Apr 18, 2019