https://file.aiquickdraw.com/mathbot/file/1727363317853muqe07qv.jpg

Let's calculate the volume of each shape step by step based on the image you've uploaded.

### 1. **Volume of the Cylinder (First Shape)**

The formula for the volume of a cylinder is:

\[
V = \pi r^2 h
\]

Where:
- \( r \) is the radius of the base.
- \( h \) is the height of the cylinder.

From the diagram:
- The diameter is \( 8 \, \text{cm} \), so the radius \( r = \frac{8}{2} = 4 \, \text{cm} \).
- The height \( h = 10 \, \text{cm} \).

Now, substitute the values:

\[
V = \pi (4)^2 (10) = \pi (16)(10) = 160\pi \, \text{cm}^3
\]

\[
V \approx 502.65 \, \text{cm}^3
\]

### 2. **Volume of the Semicircular Prism (Second Shape)**

The formula for the volume of a semicircular prism is:

\[
V = \frac{1}{2} \pi r^2 L
\]

Where:
- \( r \) is the radius of the semicircular base.
- \( L \) is the length of the prism.

From the diagram:
- The diameter of the semicircular base is \( 6 \, \text{cm} \), so the radius \( r = \frac{6}{2} = 3 \, \text{cm} \).
- The length \( L = 13 \, \text{cm} \).

Now, substitute the values:

\[
V = \frac{1}{2} \pi (3)^2 (13) = \frac{1}{2} \pi (9)(13) = \frac{117 \pi}{2} \, \text{cm}^3
\]

\[
V \approx 183.35 \, \text{cm}^3
\]

### 3. **Volume of the Solid with Semicircular End (Third Shape)**

This is a combination of a rectangular prism and a half-cylinder on top of it. So, the total volume is the sum of the volumes of the rectangular prism and the half-cylinder.

#### Rectangular Prism:
The formula for the volume of a rectangular prism is:

\[
V = L \times W \times H
\]

Where:
- \( L = 40 \, \text{cm} \), \( W = 20 \, \text{cm} \), and \( H = 30 \, \text{cm} \) are the dimensions.

So,

\[
V_{\text{rectangular prism}} = 40 \times 20 \times 30 = 24,000 \, \text{cm}^3
\]

#### Half-Cylinder:
The formula for the volume of a half-cylinder is:

\[
V = \frac{1}{2} \pi r^2 h
\]

Where:
- The diameter is \( 20 \, \text{cm} \), so the radius \( r = \frac{20}{2} = 10 \, \text{cm} \).
- The height \( h = 40 \, \text{cm} \).

Now, substitute the values:

\[
V_{\text{half-cylinder}} = \frac{1}{2} \pi (10)^2 (40) = \frac{1}{2} \pi (100)(40) = 2000\pi \, \text{cm}^3
\]

\[
V_{\text{half-cylinder}} \approx 6,283.19 \, \text{cm}^3
\]

#### Total Volume:

\[
V_{\text{total}} = 24,000 + 6,283.19 = 30,283.19 \, \text{cm}^3
\]

### 4. **Volume of the Torus (Fourth Shape)**

The formula for the volume of a torus is:

\[
V = 2 \pi^2 R r^2
\]

Where:
- \( R \) is the distance from the center of the torus to the center of the tube.
- \( r \) is the radius of the tube.

From the diagram:
- \( R = \frac{32 + 14}{2} = 23 \, \text{cm} \) (this is the average of the outer and inner diameters).
- \( r = 9 \, \text{cm} \) (radius of the tube).

Now, substitute the values:

\[
V = 2 \pi^2 (23) (9)^2 = 2 \pi^2 (23)(81) = 2 \pi^2 (1,863)
\]

\[
V \approx 36,831.99 \, \text{cm}^3
\]

### Final Volumes:
1. **Cylinder:** \( 502.65 \, \text{cm}^3 \)
2. **Semicircular Prism:** \( 183.35 \, \text{cm}^3 \)
3. **Solid with Semicircular End:** \( 30,283.19 \, \text{cm}^3 \)
4. **Torus:** \( 36,831.99 \, \text{cm}^3 \)

Would you like more details on any step or further clarifications?

#### Related Questions:
1. How do you calculate the surface area of these shapes?
2. What is the difference between calculating the volume of a full cylinder and a half-cylinder?
3. How is the formula for the volume of a torus derived?
4. How do you find the surface area of a torus?
5. How would the volume change if the dimensions of the semicircular prism were doubled?

#### Tip:
When solving for the volume of composite shapes, always break them into simpler shapes (cylinder, prism, etc.) to make calculations easier.

This detailed guide provides step-by-step volume calculations for a cylinder, semicircular prism, solid with a semicircular end, and torus. Explore the key mathematical concepts, formulas, and theorems used in these calculations. Suitable for students in Grades 9-12.

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