A storage tank for butane gas is to be built in the shape of

A storage tank for butane gas is to be built in the shape of a right circular cylinder of altitude 8 ft, with a half sphere attached to each end. If x represents the radius of each half sphere, what radius should be used to cause the volume of the tank to be 108ft^3

Solution

Volume for a right circular cylinder =
V = ()(r^2)(h) where
V = right circular cylinder volume
= the constant pi
r = radius
h = height or altitude

With a hemisphere on each end, if I calculate the volume of a sphere, that will include both hemispheres. So the volume of a sphere =
V = (4/3)()(r^3) where
V = sphere volume
= the constant pi
r = radius

So the total volume of the entire propane gas storage tank =
Vt = volume of cylinder + 2(volume of hemishere)
Vt = volume of cylinder + volume of sphere
Vt = ()(r^2)(h) + (4/3)()(r^3)
108 = ()(r^2)(8) + (4/3)()(r^3)
Divide both sides by to eliminate it.
108 = 8r^2 + (4r^3)/3
Multiply both sides of the equal sign by 3 to eliminate the denominator.
324 = 24r^2 + 4r^3
Factor a common 4 from the right side of the equal sign.
4*81 = 4(6r^2 + r^3)
81 = (6r^2 + r^3)
r = 3