To find the surface area (S.A.) of a triangular prism, we need to calculate both the lateral surface area and the areas of the triangular bases.
Step 1: Find the area of the triangular base
The base of the prism is a triangle with sides of length \( s1 = 10 \) inches, \( s2 = 12 \) inches, and \( s3 = 17 \) inches. We can use Heron's formula to find the area of the triangle.
-
Calculate the semi-perimeter \( s \): \[ s = \frac{s1 + s2 + s3}{2} = \frac{10 + 12 + 17}{2} = \frac{39}{2} = 19.5 \text{ inches} \]
-
Apply Heron's formula to find the area \( A \) of the triangle: \[ A = \sqrt{s \cdot (s - s1) \cdot (s - s2) \cdot (s - s3)} \] Thus, substituting the values we have: \[ A = \sqrt{19.5 \cdot (19.5 - 10) \cdot (19.5 - 12) \cdot (19.5 - 17)} \] \[ A = \sqrt{19.5 \cdot 9.5 \cdot 7.5 \cdot 2.5} \] Now calculating each of these: \[ 19.5 - 10 = 9.5, \quad 19.5 - 12 = 7.5, \quad 19.5 - 17 = 2.5 \] Calculating further: \[ A = \sqrt{19.5 \cdot 9.5 \cdot 7.5 \cdot 2.5} \]
Let's do the multiplication: \[ 19.5 \cdot 9.5 = 185.25, \quad 7.5 \cdot 2.5 = 18.75 \] Now multiply: \[ 185.25 \cdot 18.75 = 3473.4375 \] Taking the square root: \[ A \approx \sqrt{3473.4375} \approx 59.0 \text{ inches}^2 \]
Step 2: Calculate the lateral surface area
The lateral surface area (L.S.A.) of a triangular prism can be calculated using the formula: \[ \text{L.S.A.} = (s1 + s2 + s3) \cdot l \] Where \( l = 10 \) inches is the length of the prism.
Calculating: \[ \text{L.S.A.} = (10 + 12 + 17) \cdot 10 = 39 \cdot 10 = 390 \text{ inches}^2 \]
Step 3: Calculate the total surface area
The total surface area (S.A.) of the prism is then: \[ \text{S.A.} = 2A + \text{L.S.A.} \] Substituting the values we found: \[ \text{S.A.} = 2 \cdot 59.0 + 390 = 118 + 390 = 508 \text{ inches}^2 \]
Conclusion
Thus, the surface area of the display showcase is: \[ \boxed{508} \text{ inches}^2 \]