To find the surface area of the composite figure made of a rectangular prism and a right triangular prism, we will first calculate the surface areas of both individual prisms and then combine them while accounting for any overlapping areas.
1. Surface Area of the Rectangular Prism
The formula for the surface area of a rectangular prism is given by:
\[ SA = 2lw + 2lh + 2wh \]
where:
- \(l = 12\) ft (length)
- \(w = 8\) ft (width)
- \(h = 8\) ft (height)
Calculating the surface area:
\[ SA_{rect} = 2(12 \times 8) + 2(12 \times 8) + 2(8 \times 8) \]
Calculating each term:
- \(2(12 \times 8) = 2(96) = 192\)
- \(2(12 \times 8) = 192\)
- \(2(8 \times 8) = 2(64) = 128\)
Now, adding them up:
\[ SA_{rect} = 192 + 192 + 128 = 512 \quad \text{sq ft} \]
2. Surface Area of the Right Triangular Prism
First, find the dimensions of the triangular base:
- The width of the right triangular prism is \(8\) ft (base width).
- The height of the triangular face is derived from the Pythagorean theorem along with the information given about the hypotenuse.
Let h_t be the height of the triangular face of the prism. The right triangle has:
- one leg as \(h_t\) (unknown),
- the other leg as \(8\) ft,
- hypotenuse as \(15\) ft.
From the Pythagorean theorem, we have:
\[ h_t^2 + 8^2 = 15^2 \] \[ h_t^2 + 64 = 225 \] \[ h_t^2 = 225 - 64 = 161 \] \[ h_t = \sqrt{161} \approx 12.688 \quad \text{ft} \]
The surface area of the right triangular prism can be calculated using the formula:
\[ SA_{triangular\ prism} = (base\ area) \times height + (perimeter \times height) \]
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Calculate the area of the triangular base: \[ \text{Area}_{triangular\ base} = \frac{1}{2} \times base \times height = \frac{1}{2} \times 8 \times h_t = \frac{1}{2} \times 8 \times \sqrt{161} \approx 50.96 \quad \text{sq ft} \]
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Calculate the perimeter of the triangular base:
- The sides are \(8\), \(15\), and \(h_t\). \[ \text{Perimeter} = 8 + 15 + \sqrt{161} \approx 8 + 15 + 12.688 \approx 35.688 \quad \text{ft} \]
- Compute the total surface area: Height \(h_{triangular\ prism} = 12\) ft (same as the height of the rectangular prism).
The lateral surface area is computed as follows: \[ SA_{triangular\ prism} = \text{Area}_{triangular\ base} + (Perimeter \times h) \] \[ = (50.96) + (35.688 \times 12) \] \[ = 50.96 + 428.256 \approx 479.216 \quad \text{sq ft} \]
3. Combine the Surface Areas
Total surface area is:
\[ SA_{total} = SA_{rect} + SA_{triangular\ prism} \]
However, we must subtract the area of the top face of the rectangular prism, which is overlapping with the base of the triangular prism.
Area of the overlapping face \(= 12 \times 8 = 96\) sq ft.
So the final calculation will be:
\[ SA_{total} = 512 + 479.216 - 96 \] \[ = 512 + 479.216 - 96 \] \[ = 895.216 \quad \text{sq ft} \]
Thus, the surface area of the composite figure is approximately:
\[ \boxed{895.22 \text{ ft}^2} \]