Constructing a storage tank A storage tank for propane gas is to be constructed in the sha...

Constructing a storage tank A storage tank for propane gas is to be constructed in the shape of a right circular cylinder of altitude 10 feet with a hemisphere attached to each end. Determine the radius x so that the resulting volume is 27 π ft3. (See Example.)

EXAMPLE

Expressing the volume of a tank as a function of its radius

A steel storage tank for propane gas is to be constructed in the shape of a right circular cylinder of altitude 10 feet with a hemisphere attached to each end. The radius r is yet to be determined. Express the volume V (in ft3) of the tank as a function of r (in feet).

SOLUTION The tank is illustrated in Figure. We may find the volume of the cylindrical part of the tank by multiplying the altitude 10 by the area πr2 of the base of the cylinder. This gives us

The two hemispherical ends, taken together, form a sphere of radius r. Using the formula for the volume of a sphere, we obtain

Thus, the volume V of the tank is

This formula expresses V as a function of r. In factored form,

FIGURE 12