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

Answer 1

To find the surface area of a triangular prism, we need to consider both the lateral surface area and the area of the two triangular bases.

### Step 1: Calculate the area of the triangular base
For a triangle with sides \(s_1\), \(s_2\), and \(s_3\), we can use Heron's formula to find the area.

1. First, calculate the semi-perimeter \(s\):
\[
s = \frac{s_1 + s_2 + s_3}{2} = \frac{10 + 12 + 17}{2} = 19.5 \text{ inches}
\]

2. Now apply Heron's formula:
\[
A = \sqrt{s(s - s_1)(s - s_2)(s - s_3)}
\]
\[
A = \sqrt{19.5(19.5 - 10)(19.5 - 12)(19.5 - 17)}
\]
\[
A = \sqrt{19.5(9.5)(7.5)(2.5)}
\]
\[
= \sqrt{19.5 \times 9.5 \times 7.5 \times 2.5}
\]

Calculating this step-by-step:

- \(19.5 \times 9.5 = 185.25\)
- \(7.5 \times 2.5 = 18.75\)
- \(185.25 \times 18.75 = 3473.4375\)
- Finally, taking the square root:
\[
A \approx 59.0 \text{ square inches}
\]

### Step 2: Calculate the lateral surface area
The lateral surface area (LSA) of a triangular prism can be calculated as:
\[
LSA = (s_1 + s_2 + s_3) \times l
\]
- Substituting the values:
\[
LSA = (10 + 12 + 17) \times 10 = 39 \times 10 = 390 \text{ square inches}
\]

### Step 3: Calculate the total surface area
The total surface area (S.A.) of the triangular prism is given by:
\[
S.A. = LSA + 2 \times A
\]
- Substituting the values we have:
\[
S.A. = 390 + 2 \times 59 \approx 390 + 118 = 508 \text{ square inches}
\]

### Final Answer:
The surface area of the display showcase is approximately \( \boxed{508} \) square inches.