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.

Answer 1

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

### Step 1: Calculate the Area of the Triangular Base

The area \(A\) of the triangular base can be calculated using Heron's formula, which states:

\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]

where \(s\) is the semi-perimeter:

\[
s = \frac{a+b+c}{2}
\]

In this case, let's denote the sides of the triangle as \(s1\), \(s2\), and \(s3\):

- \(s1 = 10\) inches
- \(s2 = 12\) inches
- \(s3 = 17\) inches

First, calculate the semi-perimeter \(s\):

\[
s = \frac{10 + 12 + 17}{2} = \frac{39}{2} = 19.5 \text{ inches}
\]

Now, we can use Heron's formula to find the area \(A\):

\[
A = \sqrt{19.5(19.5 - 10)(19.5 - 12)(19.5 - 17)}
\]

Calculating each term:

\[
19.5 - 10 = 9.5
\]
\[
19.5 - 12 = 7.5
\]
\[
19.5 - 17 = 2.5
\]

Now substituting these values into the area formula:

\[
A = \sqrt{19.5 \times 9.5 \times 7.5 \times 2.5}
\]

Calculating the product step by step:

1. \(19.5 \times 9.5 = 185.25\)
2. \(7.5 \times 2.5 = 18.75\)

Now multiply these two results:

\[
185.25 \times 18.75 \approx 3472.96875
\]

Now take the square root:

\[
A \approx \sqrt{3472.96875} \approx 58.904 \text{ square inches}
\]

### Step 2: Calculate the Lateral Surface Area

The lateral surface area of a prism can be calculated using the formula:

\[
\text{Lateral Surface Area} = (\text{perimeter of the base}) \times \text{height}
\]

The perimeter \(P\) of the triangular base is:

\[
P = s1 + s2 + s3 = 10 + 12 + 17 = 39 \text{ inches}
\]

Now, multiply by the height of the prism:

\[
\text{Lateral Surface Area} = P \times h = 39 \times 9 = 351 \text{ square inches}
\]

### Step 3: Calculate the Total Surface Area

Finally, the total surface area \(SA\) of the prism is given by:

\[
SA = 2A + \text{Lateral Surface Area}
\]

Substituting in our values:

\[
SA = 2 \times 58.904 + 351
\]
\[
SA \approx 117.808 + 351 \approx 468.808 \text{ square inches}
\]

### Final Answer

Thus, the surface area of the display showcase is approximately:

\[
\boxed{468.81} \text{ square inches}
\]