Use the shell method to find the volume of the solid formed when a hole of radius 6 is drilled symmetrically along the axis of a right circular cone of radius 8 and height 12 .
Real Tutor Solution
Answer
Solution
Reviewed and approved by the UpStudy tutoring team
Explain 
Simplify this solution Beyond the Answer
To find the volume of the solid formed by a right circular cone with a cylindrical hole drilled through it, we start by determining the volume of the cone and then subtracting the volume of the cylindrical hole. The cone's volume \( V_{\text{cone}} \) can be calculated using the formula: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] where \( r = 8 \) (radius of the cone) and \( h = 12 \) (height of the cone). Plugging in the values gives: \[ V_{\text{cone}} = \frac{1}{3} \pi (8^2)(12) = \frac{1}{3} \pi (64)(12) = \frac{768}{3} \pi = 256 \pi \] Next, we find the volume of the cylindrical hole \( V_{\text{cylinder}} \), which has a radius of 6 and a height equal to the height of the cone, 12: \[ V_{\text{cylinder}} = \pi r^2 h = \pi (6^2)(12) = \pi (36)(12) = 432 \pi \] Finally, we subtract the volume of the cylinder from the volume of the cone: \[ V_{\text{solid}} = V_{\text{cone}} - V_{\text{cylinder}} = 256 \pi - 432 \pi = -176 \pi \] However, a negative volume doesn't make sense in physical terms, which indicates that the radius of the hole is too large to fit within the dimensions of the cone. Therefore, in order for the hole to be drilled through the cone, the radius of the hole must be smaller than the dimensions of the cone. Thus, the procedure demonstrates the importance of ensuring that geometric constraints are observed: always check that any drilled hole fits within the original solid before attempting to calculate the volume.
