To find the surface area of the composite figure, we need to calculate the surface areas of both the rectangular prism and the right triangular prism, then combine them while accounting for any areas that are not exposed (the areas that are in contact).
Step 1: Surface Area of the Rectangular Prism
The rectangular prism has:
- Length = 12 feet
- Width = 8 feet
- Height = 8 feet
The surface area \(SA\) of a rectangular prism is given by the formula: \[ SA = 2(lw + lh + wh) \] Where:
- \(l\) = length
- \(w\) = width
- \(h\) = height
Substituting in our values: \[ SA = 2(12 \times 8 + 12 \times 8 + 8 \times 8) \] Calculating each term:
- \(12 \times 8 = 96\)
- \(12 \times 8 = 96\)
- \(8 \times 8 = 64\)
Now substituting these values back into the equation: \[ SA = 2(96 + 96 + 64) = 2(256) = 512 \text{ sq ft} \]
Step 2: Surface Area of the Right Triangular Prism
The right triangular prism has:
- Width (base) = 8 feet
- Height (height of the triangular face) = \(h\) (height of the prism is remaining after accounting for the height of the rectangular prism)
- Hypotenuse = 15 feet
The height of the triangular face can be calculated using the Pythagorean theorem, where \(b = 8\) and we'll solve for \(h\): If the triangular base is 8 feet, and the hypotenuse is 15 feet, we can find the height \(h\) as follows:
Let the base of the triangle \(b\) = 8 feet, and use the formula: \[ b^2 + h^2 = \text{hypotenuse}^2 \Rightarrow 8^2 + h^2 = 15^2 \] Calculating the squares: \[ 64 + h^2 = 225 \] Solving for \(h^2\): \[ h^2 = 225 - 64 = 161 \] Taking the square root to find \(h\): \[ h = \sqrt{161} \approx 12.6885 \text{ feet} \]
Now calculating the surface area of the triangular prism:
The surface area \(SA\) of the triangular prism is given by: \[ SA = bh + \text{perimeter of the triangular base} \times \text{height} \] Where:
- The base area \( = \frac{1}{2} \times \text{base} \times \text{height}\)
- The perimeter of the triangular base is \(b + h + \text{hypotenuse}\)
Calculating the area of the base: \[ \text{Area} = \frac{1}{2} \times 8 \times \sqrt{161} \approx 0.5 \times 8 \times 12.6885 \approx 50.75 \text{ sq ft} \]
Now the perimeter of the triangle: \[ \text{Perimeter} = 8 + \sqrt{161} + 15 = 23 + \sqrt{161} \approx 30.6885 \]
The height of the triangular prism also equals the height of the rectangular prism (8 feet). Thus, the lateral area contribution of the triangular prism is: \[ \text{Lateral Surface Area} = \text{Perimeter} \times \text{height} \approx 30.6885 \times 8 \approx 245.508 \]
So the total surface area of the triangular prism is: \[ SA = \text{Area of base} + \text{Lateral Area} \approx 50.75 + 245.508 \approx 296.258 \]
Step 3: Total Surface Area of the Composite Figure
Now we sum the areas of both prisms: \[ SA_{\text{total}} = SA_{\text{rectangular prism}} + SA_{\text{triangular prism}} \approx 512 + 296.258 \approx 808.258 \]
However, because the triangular prism sits on top of the rectangular prism, the area of the top face of the rectangular prism (which is \(12 \times 8 = 96\) sq ft) is not exposed. Thus, we need to subtract this area from our total:
\[ SA_{\text{final}} = 808.258 - 96 \approx 712.258 \]
Conclusion
The surface area of the composite figure is approximately \(712.3 , \text{sq ft}\) when rounded appropriately, but if only integers are accepted, you could potentially report it as: \[ \boxed{712} \text{ sq ft} \]