Question

[20 POINTS] A solid machine part is to be manufactured as shown in the figure. The part is made by cutting a small cone off the top of a larger cone. The small cone has a base radius of 3 inches and a height of 5 inches. The larger cone has a base radius of 9 inches and had a height of 15 inches prior to being cut. What is the volume of the resulting part illustrated in the figure? A solid machine part has a conical shape with a cut on its apex, formed by cutting a small cone on the apex of a larger cone.

Answer 1

To find the volume of the resulting part, we need to find the volumes of the larger cone and the smaller cone, and then subtract the volume of the smaller cone from the volume of the larger cone.

The volume of a cone can be calculated using the formula V = (1/3)πr²h, where r is the base radius and h is the height.

For the larger cone:

Radius (r) = 9 inches

Height (h) = 15 inches

V = (1/3)π * 9² * 15
V = (1/3)π * 81 * 15

V = π * 27 * 15

V = 405π cubic inches

For the smaller cone:

Radius (r) = 3 inches

Height (h) = 5 inches

V = (1/3)π * 3² * 5

V = (1/3)π * 9 * 5

V = π * 3 * 5

V = 15π cubic inches

Therefore, the volume of the resulting part is:

Volume of larger cone - Volume of smaller cone= 405π - 15π= 390π cubic inches

The volume of the resulting part is 390π cubic inches.