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Part 1
Use the shell method to find the volume of the solid formed when a hole of radius 2 is drilled symmetrically along the axis of a right circular cone of radius 4 and height 6.
Question content area bottom
Part 1
Model the situation on a set of axes by placing the center of the base of the cone at the origin and the​ cone's axis along the positive​ y-axis. Set up the integral that gives the volume of the solid using the shell method

To find the volume of the solid formed when a hole of radius 2 is drilled along the axis of a right circular cone using the shell method, we'll first set up the problem based on the given coordinates.

### Step-by-Step Solution:

1. **Model the Cone and Hole:**

   - The right circular cone has a base radius of 4 and a height of 6.
   - We place the center of the base of the cone at the origin \((0, 0)\), and its axis along the positive \(y\)-axis. Therefore, the vertex of the cone is at \((0, 6)\).

2. **Find the Equation of the Cone:**

   The cone's slope is \(-\frac{\text{radius}}{\text{height}} = -\frac{4}{6} = -\frac{2}{3}\).

   Since the base is centered at the origin, the equation of the line that forms the side of the cone is:
   \[
   x = -\frac{2}{3}y + \frac{8}{3}.
   \]

3. **Shell Method Setup:**

   The shell method involves integrating along the axis perpendicular to the axis of revolution (the \(y\)-axis here).

   The volume of the solid (cone with a cylindrical hole removed) is given by:
   \[
   V = 2 \pi \int_{y=0}^{y=6} \left( \text{radius} \times \text{height of shell} \right) \, dy.
   \]

   For each shell at height \(y\):

   - The radius of the shell (distance from the y-axis) is \(x = -\frac{2}{3}y + \frac{8}{3}\).
   - The thickness of the shell is \(\,dy\).

   So, the volume of the hole-drilled shell is:
   \[ 
   V = \left( \int_{0}^{3} r\pi dy \right )
   ```

Use the shell method to find the volume of the solid formed when a hole of radius 2 is drilled symmetrically along the axis of a right circular cone of radius 4 and height 6. Model the situation on a set of axes by placing the center of the base of the cone at the origin and the​ cone's axis along the positive​ y-axis. Set up the integral that gives the volume of the solid using the shell method.

Learn how to find the volume of a solid formed when a hole of radius 2 is drilled through a cone of radius 4 and height 6 using the shell method. This calculus problem involves setting up and solving the integral to compute the volume using the cone's equation and the shell method. Suitable for college-level calculus students.

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