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 .

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 using the shell method, we can follow these steps: ### Step 1: Understand the Geometry 1. **Cone Dimensions**: - Radius \( R = 8 \) - Height \( h = 12 \) 2. **Drilled Hole**: - Radius of the hole \( r = 6 \) ### Step 2: Set Up the Problem The volume of the solid can be found by calculating the volume of the cone and subtracting the volume of the cylindrical hole. ### 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: \[ V_c = \frac{1}{3} \pi (8^2)(12) \] ### Step 4: Volume of the Cylindrical Hole The volume \( V_h \) of a cylinder is given by the formula: \[ V_h = \pi r^2 h \] The height of the cylinder is the same as the height of the cone, which is 12. Thus: \[ V_h = \pi (6^2)(12) \] ### Step 5: Calculate the Volumes Now, we will calculate both volumes and then find the volume of the solid by subtracting the volume of the hole from the volume of the cone. Let's perform the calculations for \( V_c \) and \( V_h \). 1. Calculate \( V_c \): \[ V_c = \frac{1}{3} \pi (8^2)(12) = \frac{1}{3} \pi (64)(12) \] 2. Calculate \( V_h \): \[ V_h = \pi (6^2)(12) = \pi (36)(12) \] Now, let's compute these values. Calculate the value by following steps: - step0: Calculate: \(\pi \times 6^{2}\times 12\) - step1: Multiply the terms: \(36\pi \times 12\) - step2: Multiply the terms: \(432\pi \) Calculate or simplify the expression \( (1/3) * \pi * (8^2) * 12 \). Calculate the value by following steps: - step0: Calculate: \(\frac{1}{3}\pi \times 8^{2}\times 12\) - step1: Multiply the terms: \(4\pi \times 8^{2}\) - step2: Multiply the terms: \(256\pi \) ### Step 6: Calculate the Volumes 1. **Volume of the Cone**: \[ V_c = 256\pi \] 2. **Volume of the Cylindrical Hole**: \[ V_h = 432\pi \] ### Step 7: Volume of the Solid Now, we find the volume of the solid formed by subtracting the volume of the hole from the volume of the cone: \[ V = V_c - V_h = 256\pi - 432\pi \] \[ V = -176\pi \] ### Conclusion Since the volume cannot be negative, this indicates that the cylindrical hole is larger than the cone's volume, which means that drilling the hole would remove more volume than the cone contains. Thus, the solid formed in this scenario would not exist as described. If you have any further questions or need clarification, feel free to ask!