A right angled triangle ABC of sides AB=6, BC=8 and AC=10 is spun once about AB and once about BC. What is the difference in volumes of the two solids formed?

Options

A. `24pi`
B. `32pi`
C. `64pi`
D. `96pi`

Answer

To solve the problem, we need to find the volumes of the solids formed when the right-angled triangle ABC is spun about sides AB and BC, and then calculate the difference between these two volumes.

### Step-by-Step Solution:

1. **Identify the Triangle and Its Dimensions**:
- We have a right-angled triangle ABC with sides:
- AB = 6 (height when spun around AB)
- BC = 8 (radius when spun around AB)
- AC = 10 (hypotenuse)

2. **Volume of the Solid Formed by Spinning Around AB**:
- When the triangle is spun around side AB, it forms a cone.
- The formula for the volume \( V \) of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]
- Here, the radius \( r \) is BC = 8 and the height \( h \) is AB = 6.
- Substitute the values into the formula:
\[ V_1 = \frac{1}{3} \pi (8)^2 (6) \]
- Calculate \( V_1 \):
\[ V_1 = \frac{1}{3} \pi (64)(6) = \frac{1}{3} \pi (384) = 128\pi \]

3. **Volume of the Solid Formed by Spinning Around BC**:
- When the triangle is spun around side BC, it also forms a cone.
- Here, the radius \( r \) is AB = 6 and the height \( h \) is BC = 8.
- Substitute the values into the formula:
\[ V_2 = \frac{1}{3} \pi (6)^2 (8) \]
- Calculate \( V_2 \):
\[ V_2 = \frac{1}{3} \pi (36)(8) = \frac{1}{3} \pi (288) = 96\pi \]

4. **Calculate the Difference in Volumes**:
- Now, we find the difference between the two volumes:
\[ \text{Difference} = V_1 - V_2 = 128\pi - 96\pi = 32\pi \]

### Final Answer:
The difference in volumes of the two solids formed is \( 32\pi \).