Question

Christian collects model cars and planes. He has a display showcase of all of his collectors' items. Solve this real-world problem to find the surface area of the display showcase if it is the shape of a triangular prism with the following dimensions: h
=9 inches, b
=17 inches, s1
=10 inches, s2
=12 inches, s3
= 17 and l
=10 inches.(1 point)
S.A.=
in.2
Remaining Attempts : 3
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Answer 1

To find the surface area of a triangular prism, we need to calculate the areas of the two triangular bases and the three rectangular lateral surfaces.

### Given:
- Height of prism \( h = 9 \) inches (which is the height of the triangle)
- Base of triangle \( b = 17 \) inches
- Sides of the triangle \( s_1 = 10 \) inches, \( s_2 = 12 \) inches, \( s_3 = 17 \) inches
- Length of the prism \( l = 10 \) inches

### Step 1: Area of the triangular base
To find the area of the triangle, we can use the formula:
\[ A = \frac{1}{2} \times base \times height \]
Using the height and the base of the triangle:
\[ A = \frac{1}{2} \times 17 \times 9 \]
\[ A = \frac{1}{2} \times 153 \]
\[ A = 76.5 \, \text{square inches} \]

### Step 2: Area of the two triangular bases
Since there are two triangular bases, the combined area is:
\[ A_{total\,triangles} = 2 \times A = 2 \times 76.5 = 153 \, \text{square inches} \]

### Step 3: Area of the rectangular lateral surfaces
Now, we calculate the area of the three rectangles (lateral surfaces) formed by each side of the triangle and the length of the prism.

1. For side \( s_1 = 10 \):
\[ A_1 = s_1 \times l = 10 \times 10 = 100 \, \text{square inches} \]

2. For side \( s_2 = 12 \):
\[ A_2 = s_2 \times l = 12 \times 10 = 120 \, \text{square inches} \]

3. For side \( s_3 = 17 \):
\[ A_3 = s_3 \times l = 17 \times 10 = 170 \, \text{square inches} \]

### Step 4: Total area of the lateral surfaces
\[ A_{total\,rectangles} = A_1 + A_2 + A_3 = 100 + 120 + 170 = 390 \, \text{square inches} \]

### Step 5: Total surface area of the prism
Now, we add the area of the triangular bases and the lateral surfaces together:
\[ S.A. = A_{total\,triangles} + A_{total\,rectangles} = 153 + 390 = 543 \, \text{square inches} \]

### Final answer:
\[ S.A. = 543 \, \text{in}^2 \]