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 \(a+b\).
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