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:
- \(19.5 \times 9.5 = 185.25\)
- \(7.5 \times 2.5 = 18.75\)
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:
\[ \boxed{468.81} \text{ square inches} \]