The roof of a gazebo is shaped like an octagonal prism. The base of the roof is a regular octagon, with each side measuring 4 meters. The height of the prism is 3 meters. A contractor is tasked with covering the entire roof (both the octagonal bases and the sides) with metal sheeting. If the cost of the metal sheeting is $50 per square meter, what will be the total cost to cover the roof?

To calculate the total cost of covering the gazebo's roof with metal sheeting, we need to first determine the surface area of the entire roof. Since the roof is shaped like an octagonal prism, we will calculate the surface area of both the **octagonal bases** and the **sides** (the lateral area of the prism). Then, we can multiply the total surface area by the cost per square meter of the metal sheeting.

### Step 1: Surface Area of the Octagonal Base
Each base is a regular octagon with a side length of 4 meters. To calculate the area $ A_{\text{base}} $ of a regular octagon, we can use the following formula:

\[
A_{\text{base}} = 2 \cdot (1 + \sqrt{2}) \cdot s^2
\]
where:
- $ s = 4 $ meters is the side length.

Plugging in the value of $ s $:

\[
A_{\text{base}} = 2 \cdot (1 + \sqrt{2}) \cdot 4^2 = 2 \cdot (1 + \sqrt{2}) \cdot 16
\]

\[
A_{\text{base}} = 2 \cdot (1 + 1.414) \cdot 16 = 2 \cdot 2.414 \cdot 16 = 77.25 \, \text{m}^2
\]

Each octagonal base has an area of 77.25 square meters. Since there are **two bases** (top and bottom), the total area of the bases is:

\[
\text{Total base area} = 2 \cdot 77.25 = 154.5 \, \text{m}^2
\]

### Step 2: Lateral Surface Area of the Prism
The lateral area is the area of the sides of the prism, which consists of 8 rectangular faces, one for each side of the octagon. The area of each rectangular face is given by:

\[
A_{\text{rect}} = \text{side length} \times \text{height} = 4 \, \text{m} \times 3 \, \text{m} = 12 \, \text{m}^2
\]

Since there are 8 sides, the total lateral area is:

\[
A_{\text{lateral}} = 8 \times 12 = 96 \, \text{m}^2
\]

### Step 3: Total Surface Area
The total surface area of the roof is the sum of the areas of the two octagonal bases and the lateral area:

\[
A_{\text{total}} = 154.5 \, \text{m}^2 + 96 \, \text{m}^2 = 250.5 \, \text{m}^2
\]

### Step 4: Cost of Metal Sheeting
The cost of metal sheeting is $50 per square meter. Therefore, the total cost $ C $ to cover the entire roof is:

\[
C = 250.5 \, \text{m}^2 \times 50 \, \frac{\$}{\text{m}^2} = 12,525 \, \$
\]

### Final Answer:
The total cost to cover the roof is **$12,525**.

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Do you want further details or have any questions?

Here are 5 related questions:
1. How would the total cost change if the height of the prism increased to 5 meters?
2. What would be the cost per square meter if the price of the metal sheeting increased by 20%?
3. How is the surface area affected if the side length of the octagonal base were doubled?
4. Could we use a simpler method to estimate the lateral surface area if the number of sides increased significantly?
5. How would we calculate the volume of the octagonal prism?

**Tip:** Understanding how to break down complex shapes into simple geometric components can make surface area calculations much easier.

Learn how to calculate the total cost to cover the roof of a gazebo shaped like an octagonal prism. With a side length of 4 meters and a height of 3 meters, this problem involves determining the surface area of the prism and the cost of metal sheeting. Suitable for students in Grades 9-12.

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