[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.
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.
To find the volume of the resulting part, we need to calculate the volume of the larger cone minus the volume of the smaller cone.
The volume of a cone is given by the formula:
V = (1/3) * π * r^2 * h
For the larger cone:
Radius (r1) = 9 inches
Height (h1) = 15 inches
V1 = (1/3) * π * (9^2) * 15
= (1/3) * π * 81 * 15
= 405π
For the smaller cone:
Radius (r2) = 3 inches
Height (h2) = 5 inches
V2 = (1/3) * π * (3^2) * 5
= (1/3) * π * 9 * 5
= 15π
Therefore, the volume of the resulting part is:
Volume = V1 - V2
= 405π - 15π
= 390π
Hence, the volume of the resulting part is 390π cubic inches.