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

the answer is 543 the bot is wrong

Answer 2

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

### Step 1: Area of the triangular base

Given the triangle has sides:
- \( s1 = 10 \) inches
- \( s2 = 12 \) inches
- \( s3 = 17 \) inches

We can use Heron's formula to find the area of the triangle.

1. Calculate the semi-perimeter \( s \):
\[
s = \frac{s1 + s2 + s3}{2} = \frac{10 + 12 + 17}{2} = 19.5 \text{ inches}
\]

2. Now we calculate the area \( A \) using Heron's formula:
\[
A = \sqrt{s(s - s1)(s - s2)(s - s3)}
\]
Substituting the values:
\[
A = \sqrt{19.5(19.5 - 10)(19.5 - 12)(19.5 - 17)}
\]
\[
A = \sqrt{19.5 \times 9.5 \times 7.5 \times 2.5}
\]

Calculate each term:
\[
19.5 - 10 = 9.5
\]
\[
19.5 - 12 = 7.5
\]
\[
19.5 - 17 = 2.5
\]

Now compute:
\[
19.5 \times 9.5 = 185.25
\]
\[
7.5 \times 2.5 = 18.75
\]
\[
185.25 \times 18.75 = 3483.75
\]
\[
A = \sqrt{3483.75} \approx 59.00 \text{ square inches}
\]

### Step 2: Area of the lateral surfaces

Now calculate the area of the three rectangular lateral faces.
- The height of the prism (length) is \( l = 10 \) inches.
- The dimensions of the rectangles are:
- Rectangle 1 (base \( s1 = 10 \)): \( A1 = s1 \times l = 10 \times 10 = 100 \text{ in}^2 \)
- Rectangle 2 (base \( s2 = 12 \)): \( A2 = s2 \times l = 12 \times 10 = 120 \text{ in}^2 \)
- Rectangle 3 (base \( s3 = 17 \)): \( A3 = s3 \times l = 17 \times 10 = 170 \text{ in}^2 \)

### Step 3: Total lateral surface area

Add up the lateral areas:
\[
A_{\text{lateral}} = A1 + A2 + A3 = 100 + 120 + 170 = 390 \text{ in}^2
\]

### Step 4: Total surface area of the prism

The total surface area \( S.A. \) of the triangular prism is given by:
\[
S.A. = 2 \times \text{Area of the triangle} + \text{Area of lateral faces}
\]
Substituting in the values:
\[
S.A. = 2 \times 59 + 390 = 118 + 390 = 508 \text{ in}^2
\]

### Final Answer

Therefore, the surface area of the display showcase is:
\[
\text{S.A.} = 508 \text{ in}^2
\]