A shape as shown is made from a cylinder 50 mm diameter and 100 mm long with a hemisphere on one end. What is the volume and surface area?

The given shape consists of a cylinder with a hemisphere attached to one of its ends. Let's break down the problem and calculate the **volume** and **surface area** of the entire shape.

### 1. **Volume Calculation**
- **Cylinder**:
  - Radius \(r\) = 50 mm / 2 = 25 mm
  - Height \(h\) = 100 mm
  - Volume of a cylinder, \(V_{\text{cylinder}}\) = \( \pi r^2 h \)
  
- **Hemisphere**:
  - Radius \(r\) = 25 mm (same as the radius of the cylinder)
  - Volume of a hemisphere, \(V_{\text{hemisphere}}\) = \( \frac{2}{3} \pi r^3 \)

The total volume of the shape is the sum of the volumes of the cylinder and the hemisphere.

\[
V_{\text{total}} = V_{\text{cylinder}} + V_{\text{hemisphere}} = \pi r^2 h + \frac{2}{3} \pi r^3
\]

### 2. **Surface Area Calculation**
- **Cylinder**:
  - Lateral surface area of the cylinder, \(A_{\text{lateral}}\) = \(2 \pi r h\)
  - The top face of the cylinder (excluding the hemisphere end) is a circle, so its area is:
    \[
    A_{\text{top}} = \pi r^2
    \]
  
- **Hemisphere**:
  - The curved surface area of a hemisphere (excluding the circular base attached to the cylinder) is:
    \[
    A_{\text{hemisphere}} = 2 \pi r^2
    \]
    
The total surface area is the sum of the lateral area of the cylinder, the top circular area of the cylinder, and the curved surface area of the hemisphere.

\[
A_{\text{total}} = A_{\text{lateral}} + A_{\text{top}} + A_{\text{hemisphere}} = 2 \pi r h + \pi r^2 + 2 \pi r^2
\]

Now let's calculate the values using \(r = 25\) mm and \(h = 100\) mm.

### Volume Calculation
\[
V_{\text{total}} = \pi (25)^2 (100) + \frac{2}{3} \pi (25)^3
\]
\[
V_{\text{total}} = \pi (625) (100) + \frac{2}{3} \pi (15625)
\]
\[
V_{\text{total}} = 62500 \pi + \frac{2}{3} \cdot 15625 \pi
\]
\[
V_{\text{total}} = 62500 \pi + 10416.67 \pi
\]
\[
V_{\text{total}} = 72916.67 \pi \, \text{mm}^3
\]
\[
V_{\text{total}} \approx 229,183.12 \, \text{mm}^3
\]

### Surface Area Calculation
\[
A_{\text{total}} = 2 \pi (25)(100) + \pi (25)^2 + 2 \pi (25)^2
\]
\[
A_{\text{total}} = 5000 \pi + 625 \pi + 1250 \pi
\]
\[
A_{\text{total}} = 6875 \pi \, \text{mm}^2
\]
\[
A_{\text{total}} \approx 21,606.81 \, \text{mm}^2
\]

### Final Answer:
- **Volume** ≈ 229,183.12 mm³
- **Surface Area** ≈ 21,606.81 mm²

Would you like further details on any of these steps?

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Here are 5 related questions:
1. How would the volume change if the radius of the cylinder and hemisphere were doubled?
2. What is the effect on the surface area if the height of the cylinder were halved?
3. How does the volume of a full sphere compare to that of the hemisphere?
4. Can you calculate the surface area of the hemisphere alone?
5. How would the calculations change if the cylinder were hollow?

**Tip**: When dealing with combined shapes, it's useful to break them down into simpler geometric figures (e.g., cylinder, hemisphere) for easier calculation.

Calculate the volume and surface area of a shape made from a cylinder 50 mm in diameter and 100 mm long, with a hemisphere on one end. Detailed solution includes key geometry concepts, suitable for Grades 9-12.

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