“The cone's base is one foot away from the cube and the height is 2 feet from the cube.” (apparent typo “cubes” has been corrected.)
We can presume that the cone is centered inside the cube's floor. So there is a minimum one-foot clearance on each of the four sides of the cube's base. That means that the cube is 2 feet longer than the diameter of the cone's base. (One foot on each side.)
The height of the cube is 2 feet higher than the vertex of the cone. So the height is equal to the diameter of the cone.
The volume of a cone is equal to (1/3)*pi*(the radius of the cone squared)*(the height of the cone).
If we let V=volume, r=radius, and h=height, this is written as V=(1/3)*pi*h*r^2
and recalling that the diameter = 2*radius = height, we can write this as:
V=(1/3)*pi*(2*r)*r^2 = (1/3)*pi*2*r^3 = (2/3)*pi*r^3.
We are given that the volume = 486*pi*ft^3, so 486*pi=(2/3)*pi*r^3.
(Note that the units will be ft^3, so we can cancel them out.)
Divide first by pi,
486 = (2/3)*r^3
Now, divide by (2/3), which is the same as multiplying by (3/2):
729 = 9^3 = r^3
Taking the third root of both sides,
r=9 (feet).
The cube is 2 feet longer than the diameter of the cube, so 9*2 +2 = 18 +2 = 20 feet.
Each side of the cube is 20 feet, so the area is 20 feet by 20 feet, or 400 square feet.
There are six sides to a cube. (Recall that a die – the singular form of dice – has six sides, each numbered from 1 to 6.)
So 6 times 400 = 2,400 square feet.
And that is the (total) surface area of the cube.
