To find the surface area of a triangular prism, we need to calculate the area of the two triangular bases as well as the area of the three rectangular lateral surfaces.
Step 1: Area of the triangular base
Given the triangle has sides:
- \( s1 = 10 \) inches
- \( s2 = 12 \) inches
- \( s3 = 17 \) inches
We can use Heron's formula to find the area of the triangle.
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Calculate the semi-perimeter \( s \): \[ s = \frac{s1 + s2 + s3}{2} = \frac{10 + 12 + 17}{2} = 19.5 \text{ inches} \]
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Now we calculate the area \( A \) using Heron's formula: \[ A = \sqrt{s(s - s1)(s - s2)(s - s3)} \] Substituting the values: \[ A = \sqrt{19.5(19.5 - 10)(19.5 - 12)(19.5 - 17)} \] \[ A = \sqrt{19.5 \times 9.5 \times 7.5 \times 2.5} \]
Calculate each term: \[ 19.5 - 10 = 9.5 \] \[ 19.5 - 12 = 7.5 \] \[ 19.5 - 17 = 2.5 \]
Now compute: \[ 19.5 \times 9.5 = 185.25 \] \[ 7.5 \times 2.5 = 18.75 \] \[ 185.25 \times 18.75 = 3483.75 \] \[ A = \sqrt{3483.75} \approx 59.00 \text{ square inches} \]
Step 2: Area of the lateral surfaces
Now calculate the area of the three rectangular lateral faces.
- The height of the prism (length) is \( l = 10 \) inches.
- The dimensions of the rectangles are:
- Rectangle 1 (base \( s1 = 10 \)): \( A1 = s1 \times l = 10 \times 10 = 100 \text{ in}^2 \)
- Rectangle 2 (base \( s2 = 12 \)): \( A2 = s2 \times l = 12 \times 10 = 120 \text{ in}^2 \)
- Rectangle 3 (base \( s3 = 17 \)): \( A3 = s3 \times l = 17 \times 10 = 170 \text{ in}^2 \)
Step 3: Total lateral surface area
Add up the lateral areas: \[ A_{\text{lateral}} = A1 + A2 + A3 = 100 + 120 + 170 = 390 \text{ in}^2 \]
Step 4: Total surface area of the prism
The total surface area \( S.A. \) of the triangular prism is given by: \[ S.A. = 2 \times \text{Area of the triangle} + \text{Area of lateral faces} \] Substituting in the values: \[ S.A. = 2 \times 59 + 390 = 118 + 390 = 508 \text{ in}^2 \]
Final Answer
Therefore, the surface area of the display showcase is: \[ \text{S.A.} = 508 \text{ in}^2 \]