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The problem in the image asks to find the volume of a composite solid made up of a hemisphere, a right circular cylinder, and a right circular cone. The dimensions given in the diagram are:
- The radius of the base (shared by the hemisphere, cylinder, and cone) is 8 cm.
- The height of the cone is 5 cm.
- The total height of the composite solid is 10 cm.

### Step-by-step Solution:

#### 1. **Hemisphere Volume:**
The volume \( V_{\text{hemisphere}} \) of a hemisphere is given by:
\[
V_{\text{hemisphere}} = \frac{2}{3} \pi r^3
\]
where \( r = 8 \, \text{cm} \) is the radius.
\[
V_{\text{hemisphere}} = \frac{2}{3} \pi (8)^3 = \frac{2}{3} \pi (512) = \frac{1024}{3} \pi \, \text{cm}^3
\]

#### 2. **Cylinder Volume:**
The height of the cylinder is found by subtracting the height of the cone (5 cm) from the total height (10 cm):
\[
h_{\text{cylinder}} = 10 \, \text{cm} - 5 \, \text{cm} = 5 \, \text{cm}
\]
The volume \( V_{\text{cylinder}} \) of a cylinder is:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
where \( r = 8 \, \text{cm} \) and \( h = 5 \, \text{cm} \):
\[
V_{\text{cylinder}} = \pi (8)^2 (5) = \pi (64)(5) = 320 \pi \, \text{cm}^3
\]

#### 3. **Cone Volume:**
The volume \( V_{\text{cone}} \) of a right circular cone is:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
where \( r = 8 \, \text{cm} \) and \( h = 5 \, \text{cm} \):
\[
V_{\text{cone}} = \frac{1}{3} \pi (8)^2 (5) = \frac{1}{3} \pi (64)(5) = \frac{320}{3} \pi \, \text{cm}^3
\]

#### 4. **Total Volume:**
The total volume of the solid is the sum of the volumes of the hemisphere, the cylinder, and the cone:
\[
V_{\text{total}} = V_{\text{hemisphere}} + V_{\text{cylinder}} + V_{\text{cone}}
\]
Substitute the values:
\[
V_{\text{total}} = \frac{1024}{3} \pi + 320 \pi + \frac{320}{3} \pi
\]
To simplify, find a common denominator:
\[
V_{\text{total}} = \frac{1024 \pi + 960 \pi + 320 \pi}{3} = \frac{2304 \pi}{3} = 768 \pi \, \text{cm}^3
\]

Thus, the total volume of the solid is:
\[
V_{\text{total}} = 768 \pi \, \text{cm}^3 \approx 2413.72 \, \text{cm}^3
\]

Would you like more details or have any questions?

### Related questions:
1. How would the volume change if the radius of the base were increased?
2. What is the surface area of the composite solid?
3. How do the individual volumes compare to the total volume?
4. How can you compute the volume if the shape was rotated differently?
5. What is the volume if the cone's height is doubled?

### Tip:
When dealing with composite shapes, break them down into simpler components (like spheres, cones, and cylinders) and sum their volumes or surface areas accordingly.

A solid is made up of a hemisphere, a right circular cylinder, and a right circular cone as shown. Find the volume of the solid.

This problem involves calculating the volume of a composite solid made up of a hemisphere, right circular cylinder, and cone. Learn how to use geometry formulas like the volume of a hemisphere, cylinder, and cone to solve this problem, with detailed steps and final volume.

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