A ball of radius 17 has a round hole of radius 9 drilled through its center. Find the volume of the resulting solid.

To find the volume of the resulting solid after a hole of radius 9 is drilled through the center of a ball of radius 17, we can approach the problem as follows:

### Step 1: Volume of the original sphere
The volume \( V_{\text{sphere}} \) of the original sphere is given by the formula:
\[
V_{\text{sphere}} = \frac{4}{3} \pi R^3
\]
where \( R = 17 \) is the radius of the sphere.

\[
V_{\text{sphere}} = \frac{4}{3} \pi (17)^3
\]

### Step 2: Volume of the cylindrical hole
The volume of the cylindrical hole with a radius \( r = 9 \) drilled through the sphere is not as straightforward to calculate because the hole is a cylindrical hole through a sphere. However, by using the principle of Pappus's centroid theorem, we can calculate the volume removed.

When the hole is drilled through the sphere, the cylindrical portion of the sphere removed can be considered as the volume of a solid of revolution.

The volume of the removed material can be calculated by the formula:
\[
V_{\text{hole}} = 2 \pi r^2 \times \text{height}
\]
Where the height is determined by the length of the cylindrical hole, which corresponds to the length of the chord passing through the sphere.

### Step 3: Calculate the volume of the solid
The volume of the remaining solid is the difference between the volume of the original sphere and the volume of the cylindrical hole removed:

\[
V_{\text{remaining}} = V_{\text{sphere}} - V_{\text{hole}}
\]

For this problem, the specific formula simplifies due to the symmetry involved, and we do not need to directly subtract the cylindrical volume. Instead, the volume of the remaining solid is given by the expression:

\[
V_{\text{remaining}} = 2\pi \int_{r}^{R} y^2 \, dx
\]
After solving, we get:

\[
V_{\text{remaining}} = 4\pi \left(\frac{R^3}{3} - \frac{r^3}{3}\right)
\]
Substituting \( R = 17 \) and \( r = 9 \):

\[
V_{\text{remaining}} = \frac{4\pi}{3} \left( 17^3 - 9^3 \right)
\]

### Final Calculation
Let's compute the final value:

\[
V_{\text{remaining}} = \frac{4\pi}{3} \left( 4913 - 729 \right) = \frac{4\pi}{3} \times 4184
\]
\[
V_{\text{remaining}} = \frac{16736\pi}{3} \approx 17565.79 \, \text{cubic units}
\]

So, the volume of the resulting solid is approximately **17,565.79 cubic units**.

Would you like more details or have any questions?

### Relative Questions
1. What would be the volume if the hole's radius was increased to 10 units?
2. How does the location of the hole affect the resulting volume?
3. Can this problem be solved using a different method, such as slicing?
4. What happens to the volume if the radius of the sphere is decreased?
5. How does the result change if the hole is off-center?

### Tip
When dealing with the volume of solids involving symmetry, understanding and applying geometric principles like Pappus's theorem can greatly simplify the calculations.

Learn how to calculate the volume of a sphere with a radius of 17 units after drilling a cylindrical hole with a radius of 9 units through its center. This problem explores concepts of geometry, solid of revolution, and volume calculation, suitable for students in Grades 11-12.

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