Find the volume of a cone of height 8 centimeters and base radius 6 centimeters. This cone is sliced by a plane that is parallel to the base and 2 centimeters from it. Find the volumes of the two resulting solids. One is a cone, while the other is called a frustrum.
the cones are similar, with the smaller altitude 6/8 = 3/4 the altitude of the larger
So the volume of the smaller is (3/4)^3 times that of the larger
The larger cone has volume 1/3 π * 6^2 * 8 = 96π
The smaller cone has volume 27/64 * 96π = 81π/2
The frustrum has volume 96π - 81π/2 =111π/2
To find the volumes of the two resulting solids, we'll first calculate the volume of the original cone and then use that information to find the volume of each part.
1. Volume of the Original Cone:
The formula to find the volume of a cone is given by V = (1/3)πr^2h, where V represents the volume, π is a constant approximately equal to 3.14159, r is the base radius, and h is the height.
Plugging in the given values, we have:
V = (1/3)π(6^2)(8)
V = (1/3)π(36)(8)
V = (1/3)(3.14159)(36)(8)
V ≈ 301.5926 cubic centimeters
2. Volume of the Smaller Cone:
The smaller cone is formed by slicing the original cone with a plane parallel to the base, 2 centimeters from it. The height of the smaller cone will also be 8 centimeters since the slicing plane is parallel to the base. However, the radius will be reduced to (6 - 2) = 4 centimeters.
Using the formula V = (1/3)πr^2h, we can calculate the volume of the smaller cone:
V_small = (1/3)π(4^2)(8)
V_small = (1/3)π(16)(8)
V_small = (1/3)(3.14159)(16)(8)
V_small ≈ 134.0413 cubic centimeters
3. Volume of the Frustrum:
The frustrum is the remaining part after slicing the original cone. It can be thought of as the original cone with the smaller cone removed. Thus, the volume of the frustrum is the difference between the volume of the original cone and the volume of the smaller cone.
V_frustrum = V - V_small
V_frustrum ≈ 301.5926 - 134.0413
V_frustrum ≈ 167.5513 cubic centimeters
So, the volume of the smaller cone is approximately 134.0413 cubic centimeters, while the volume of the frustrum is approximately 167.5513 cubic centimeters.