Use the shell method to find the volume of the solid formed when a hole of radius 1 is drilled symmetrically along the axis of a right circular cone of radius 6 and height 9 .

To find the volume of the solid formed when a hole of radius 1 is drilled symmetrically along the axis of a right circular cone of radius 6 and height 9 using the shell method, we can follow these steps: ### Step 1: Understand the Geometry 1. **Cone Dimensions**: - Radius \( R = 6 \) - Height \( H = 9 \) 2. **Drilled Hole**: - Radius of the hole \( r = 1 \) ### Step 2: Set Up the Problem The volume of the solid can be found by subtracting the volume of the cylindrical hole from the volume of the cone. ### Step 3: Volume of the Cone The volume \( V_c \) of a cone is given by the formula: \[ V_c = \frac{1}{3} \pi R^2 H \] Substituting the values: \[ V_c = \frac{1}{3} \pi (6^2)(9) \] ### Step 4: Volume of the Cylindrical Hole The volume \( V_h \) of a cylinder is given by the formula: \[ V_h = \pi r^2 h \] where \( h \) is the height of the cylinder. Since the hole goes through the entire height of the cone, \( h = 9 \): \[ V_h = \pi (1^2)(9) \] ### Step 5: Calculate the Volumes Now, we will calculate both volumes. 1. **Volume of the Cone**: \[ V_c = \frac{1}{3} \pi (6^2)(9) = \frac{1}{3} \pi (36)(9) = \frac{1}{3} \pi (324) = 108\pi \] 2. **Volume of the Cylindrical Hole**: \[ V_h = \pi (1^2)(9) = \pi (1)(9) = 9\pi \] ### Step 6: Volume of the Solid The volume \( V \) of the solid formed is: \[ V = V_c - V_h = 108\pi - 9\pi = 99\pi \] ### Final Answer Thus, the volume of the solid formed when a hole of radius 1 is drilled symmetrically along the axis of the cone is: \[ \boxed{99\pi} \]