To find the surface area of a triangular prism, we need to calculate both the lateral area and the area of the two triangular bases.
- Calculate the area of the triangular base:
The base of the prism is a triangle with a base (b) of 17 inches and height (h) of 9 inches. The area \( A \) of the triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times base \times height \]
Plugging in the values:
\[ A = \frac{1}{2} \times 17 \times 9 = \frac{153}{2} = 76.5 \text{ square inches} \]
Since the prism has two identical triangular bases, the total area for the bases is:
\[ \text{Area of bases} = 2 \times A = 2 \times 76.5 = 153 \text{ square inches} \]
- Calculate the lateral area:
The lateral area \( L \) of a prism can be found by summing the areas of the three rectangular faces. The dimensions of the rectangles are the lengths of the edges of the triangle and the height \( l \) (10 inches):
- Area of the rectangle corresponding to \( s1 \) (10 inches):
\[ L_1 = s1 \times l = 10 \times 10 = 100 \text{ square inches} \]
- Area of the rectangle corresponding to \( s2 \) (12 inches):
\[ L_2 = s2 \times l = 12 \times 10 = 120 \text{ square inches} \]
- Area of the rectangle corresponding to \( s3 \) (17 inches):
\[ L_3 = s3 \times l = 17 \times 10 = 170 \text{ square inches} \]
Now, sum these areas:
\[ L = L_1 + L_2 + L_3 = 100 + 120 + 170 = 390 \text{ square inches} \]
- Total surface area of the triangular prism:
Finally, the total surface area \( S \) of the prism is the sum of the areas of the two triangular bases and the lateral area:
\[ S = \text{Area of bases} + L = 153 + 390 = 543 \text{ square inches} \]
The surface area of the display showcase is 543 square inches.