To find the surface area (S.A.) of a triangular prism, we need to calculate the area of the two triangular bases and the three rectangular lateral faces.
- Calculate the area of the triangular base:
The formula for the area \(A\) of a triangle is:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, the base \(b = 17\) inches and the height \(h = 9\) inches. Therefore,
\[ A = \frac{1}{2} \times 17 \times 9 = \frac{153}{2} = 76.5 \text{ in}^2 \]
Since there are two triangular bases, the total area for the triangular bases is:
\[ \text{Area of the two bases} = 2 \times A = 2 \times 76.5 = 153 \text{ in}^2 \]
- Calculate the area of the rectangular faces:
The three rectangular faces have the following dimensions:
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Face 1 (length \(l = 10\) inches, height = \(s1 = 10\) inches): \[ \text{Area} = l \times s1 = 10 \times 10 = 100 \text{ in}^2 \]
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Face 2 (length \(l = 10\) inches, height = \(s2 = 12\) inches): \[ \text{Area} = l \times s2 = 10 \times 12 = 120 \text{ in}^2 \]
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Face 3 (length \(l = 10\) inches, height = \(s3 = 17\) inches): \[ \text{Area} = l \times s3 = 10 \times 17 = 170 \text{ in}^2 \]
Now, adding these areas together gives us the area of the rectangular faces:
\[ \text{Area of the three rectangular faces} = 100 + 120 + 170 = 390 \text{ in}^2 \]
- Calculate the total surface area of the prism:
Now, we can add the area of the triangular bases and the area of the rectangular faces together:
\[ \text{Total Surface Area} = \text{Area of the bases} + \text{Area of the rectangular faces} = 153 + 390 = 543 \text{ in}^2 \]
Therefore, the surface area of the display showcase is:
\[ \text{S.A.} = 543 \text{ in}^2 \]