Remember that the equation of the volume of a sphere is: [tex]V= \frac{4 \pi r^{3}}{3} [/tex]
where
[tex]V[/tex] is the volume of the sphere
[tex]r[/tex] is the radius of the sphere
We know from our problem that the volume of our spherical ball is [tex]972 \pi in^{3}[/tex], so [tex]V=972 \pi [/tex]. Lets replace that value in our equation to find [tex]r[/tex]:
[tex]V= \frac{4 \pi r^{3}}{3} [/tex]
[tex]972 \pi = \frac{4 \pi r^{3 }}{3} [/tex]
[tex]r^3= \frac{3(972 \pi )}{4 \pi } [/tex]
[tex]r^{3}=729[/tex]
[tex]r= \sqrt[3]{729} [/tex]
[tex]r=9[/tex]
Now that we have the radius of our spherical ball, we can find its diameter using the equation: [tex]d=2r[/tex]:
[tex]d=2(9)[/tex]
[tex]d=18[/tex]
Since we know for our problem that the edge length of the box is equal to the diameter of the ball, the length of one side of our square is 18 in.
Now, to find the volume of our cube, we are going to use the equation: [tex]V=a^3[/tex]
where
[tex]V[/tex] is the volume of the cube
[tex]a[/tex] is a side of the cube (remember that all the sides of a cube are equal)
[tex]V=18^3[/tex]
[tex]V=5832[/tex]
We can conclude that the volume of our cube is 5832 [tex]in^{3}[/tex].
