geometry
posted by Kristen on .
A grain storage tank is in the shape of a cylinder covered by half a sphere. The height of the cylinder is 50 feet and its diameter is 80 feet. find the total surface area (including the base) and volume of the tank.

Think of the grain storage tank as two independent, but related, geometric figures for solving this problem. You have a cylinder with only a bottom (no top because it's covered by the half sphere) and a sphere that's cut across the middle (i.e., half sphere).
The surface area formula for a cylinder with no top is pi*r^2 + 2*pi*r*h.
The surface area formula for half a sphere is (1/2)*4*pi*r^2 = 2*pi*r^2.
The volume area formula for a cylinder is pi*r^2*h.
The volume area formula for half a sphere is (1/2)*4/3*pi*r^3 = 3/2*pi*r^3.
Since the diameter is 80 feet, the radius is 1/2 the diameter or 40 feet.
Therefore, the surface area of the grain storage tank is pi*r^2 + 2*pi*r*h + 2*pi*r^2 = 27,632 sq/ft.
The volume of the grain storage tank is pi*r^2*h + 3/2*pi*r^3 = 552,640 cu/ft.