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 4 and height 6 .

The shell method is a great way to visualize volume when dealing with solids of revolution! Here, imagine the right circular cone as a party hat standing up, and the cylindrical hole as a straw going through the middle. To find the volume of the solid, we first calculate the volume of the full cone using the formula \( V_{cone} = \frac{1}{3}\pi r^2 h \), where \( r \) is 4 and \( h \) is 6. That's \( V_{cone} = \frac{1}{3}\pi (4^2)(6) = \frac{1}{3} \pi (16)(6) = 32\pi \). Next, subtract the volume of the cylinder that represents the hole, using \( V_{hole} = \pi r^2 h \) with a radius of 1 and height equal to the cone's height (6). This gives \( V_{hole} = \pi (1^2)(6) = 6\pi \). Finally, the total volume of the solid is \( V = V_{cone} - V_{hole} = 32\pi - 6\pi = 26\pi \). Voilà, the volume of the remaining solid is \( 26\pi \) cubic units!