1)A cylindrical bar of soap has a diameter of 2.5 in. and a height of 1.5 in. A cylindrical container holds a stack of three bars. What is the minimum surface area of this cylindrical container to two decimal places?

To find the minimum surface area of the cylindrical container, we need to calculate both the lateral surface area and the area of the top and bottom circular faces.

### Step 1: Determine the dimensions of the container
- **Diameter of one bar of soap**: \( d = 2.5 \) inches
- **Radius** of the container (which equals the radius of one soap bar): \( r = \frac{d}{2} = \frac{2.5}{2} = 1.25 \) inches
- **Height of the container** (which is the height of three stacked bars): \( h = 3 \times 1.5 = 4.5 \) inches

### Step 2: Calculate the lateral surface area
The lateral surface area \( A_{\text{lateral}} \) of the cylindrical container is given by:
\[
A_{\text{lateral}} = 2\pi rh
\]
Substituting the values:
\[
A_{\text{lateral}} = 2\pi(1.25)(4.5) = 11.25\pi \text{ square inches}
\]

### Step 3: Calculate the area of the top and bottom faces
The area \( A_{\text{base}} \) of each circular face is:
\[
A_{\text{base}} = \pi r^2
\]
\[
A_{\text{base}} = \pi(1.25)^2 = 1.5625\pi \text{ square inches}
\]
Since there are two circular faces (top and bottom):
\[
A_{\text{total\_base}} = 2 \times 1.5625\pi = 3.125\pi \text{ square inches}
\]

### Step 4: Calculate the total surface area
The total surface area \( A_{\text{total}} \) is the sum of the lateral surface area and the area of the top and bottom faces:
\[
A_{\text{total}} = A_{\text{lateral}} + A_{\text{total\_base}} = 11.25\pi + 3.125\pi = 14.375\pi \text{ square inches}
\]
\[
A_{\text{total}} \approx 14.375 \times 3.1416 = 45.17 \text{ square inches}
\]

### Final Answer
The minimum surface area of the cylindrical container is **45.17 square inches**.

Would you like more details, or do you have any questions about this problem?

Here are 8 related questions:
1. How would the surface area change if the height of each soap bar were increased?
2. How do you calculate the volume of the cylindrical container?
3. If the diameter of the soap bars were doubled, what would the new surface area of the container be?
4. How does the lateral surface area contribute to the total surface area of a cylinder?
5. What is the significance of the radius in determining the surface area of a cylinder?
6. How would the surface area be affected if only two bars of soap were stacked in the container?
7. Can you find the volume of one bar of soap?
8. What is the relationship between the surface area and volume of a cylinder?

**Tip:** When dealing with cylindrical objects, remember that both the height and the radius significantly influence the surface area and volume.

Learn how to calculate the minimum surface area of a cylindrical container holding three soap bars. This problem involves geometry concepts such as cylinders and surface area calculations, with detailed step-by-step solutions. Suitable for high school students.

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