To calculate the surface area of the tent-shaped prism (a triangular prism) in the provided diagram, we will follow these steps:

Key Measurements:

1. The triangular base dimensions:
   • Height: $8 \, \text{in}$
   • Base: $12 \, \text{in}$
2. The three rectangular sides:
   • Bottom rectangle: $12 \, \text{in} \times 15 \, \text{in}$
   • Two slanted rectangles: $10 \, \text{in} \times 15 \, \text{in}$ each.

Step 1: Area of the triangular base

The area of a triangle is given by: $\text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ $\text{Area of Triangle} = \frac{1}{2} \times 12 \, \text{in} \times 8 \, \text{in} = 48 \, \text{in}^2$ Each prism has two triangular bases: $2 \times 48 = 96 \, \text{in}^2$

Step 2: Areas of the three rectangles

• Bottom rectangle: $\text{Area} = 12 \, \text{in} \times 15 \, \text{in} = 180 \, \text{in}^2$
• Each slanted rectangle: $\text{Area} = 10 \, \text{in} \times 15 \, \text{in} = 150 \, \text{in}^2$ There are two slanted rectangles: $2 \times 150 = 300 \, \text{in}^2$

Step 3: Total surface area

$\text{Surface Area} = \text{Area of 2 triangles} + \text{Area of bottom rectangle} + \text{Area of slanted rectangles}$ $\text{Surface Area} = 96 + 180 + 300 = 576 \, \text{in}^2$

Final Answer:

The surface area of the prism is: $\boxed{576 \, \text{in}^2}$

Would you like me to provide additional details or clarify any part of the solution?

Here are 5 related questions to explore:

1. How is the formula for the area of a triangle derived?
2. What is the surface area formula for a cylinder?
3. How would the surface area change if the height of the triangular base increased to 10 inches?
4. How does the surface area relate to the volume of the prism?
5. What are real-world applications of calculating the surface area of a triangular prism?

Tip: To avoid errors in surface area calculations, organize the problem by separating different faces or surfaces and summing their areas systematically.