To find the surface area of a triangular prism, we need to calculate the area of the two triangular bases and the area of the three rectangular lateral faces.
The surface area \( S.A. \) of a triangular prism is given by the formula:
\[ S.A. = 2 \cdot A_{\text{triangle}} + A_{\text{lateral}} \]
Where:
- \( A_{\text{triangle}} \) is the area of the triangular base.
- \( A_{\text{lateral}} \) is the area of the lateral rectangular faces.
Step 1: Calculate the area of the triangular base
To calculate the area of the triangle, we can use Heron's formula, which requires the lengths of all three sides of the triangle.
Given:
- \( s_1 = 10 \) inches
- \( s_2 = 12 \) inches
- \( s_3 = 17 \) inches
First, we calculate the semi-perimeter \( s \):
\[ s = \frac{s_1 + s_2 + s_3}{2} = \frac{10 + 12 + 17}{2} = 19.5 \text{ inches} \]
Now, we can use Heron's formula to find the area \( A_{\text{triangle}} \):
\[ A_{\text{triangle}} = \sqrt{s(s - s_1)(s - s_2)(s - s_3)} \]
\[ A_{\text{triangle}} = \sqrt{19.5 \times (19.5 - 10) \times (19.5 - 12) \times (19.5 - 17)} \]
\[ A_{\text{triangle}} = \sqrt{19.5 \times 9.5 \times 7.5 \times 2.5} \]
Calculating the values step by step:
- \( 19.5 - 10 = 9.5 \)
- \( 19.5 - 12 = 7.5 \)
- \( 19.5 - 17 = 2.5 \)
Now multiply these numbers:
\[ 19.5 \times 9.5 \times 7.5 \times 2.5 = 19.5 \times 9.5 = 185.25 \] \[ 185.25 \times 7.5 = 1389.375 \] \[ 1389.375 \times 2.5 = 3473.4375 \]
Now find the square root:
\[ A_{\text{triangle}} = \sqrt{3473.4375} \approx 58.9 \text{ square inches} \]
Step 2: Calculate the lateral area
The lateral area consists of three rectangular faces:
- Face 1: \( \text{base} \times \text{height} = s_3 \times h = 17 \times 10 = 170 \)
- Face 2: \( s_1 \times h = 10 \times 10 = 100 \)
- Face 3: \( s_2 \times h = 12 \times 10 = 120 \)
Thus, the total lateral area is:
\[ A_{\text{lateral}} = 170 + 100 + 120 = 390 \text{ square inches} \]
Step 3: Calculate the total surface area
Now we can find the total surface area:
\[ S.A. = 2 \cdot A_{\text{triangle}} + A_{\text{lateral}} = 2 \cdot 58.9 + 390 \] \[ S.A. = 117.8 + 390 = 507.8 \text{ square inches} \]
The surface area of the display showcase is approximately:
\[ \boxed{507.8 \text{ square inches}} \]