To calculate the surface area of the composite figure made up of a rectangular prism and a right triangular prism, we will first find the surface areas of each individual prism and then combine them.
Step 1: Surface area of the rectangular prism
The formula for the surface area \(S\) of a rectangular prism is given by:
\[ S = 2lw + 2lh + 2wh \]
Where:
- \(l\) = length
- \(w\) = width
- \(h\) = height
Given:
- Length \(l = 12\) feet
- Width \(w = 8\) feet
- Height \(h = 8\) feet
Now we substitute the values:
\[ S = 2(12)(8) + 2(12)(8) + 2(8)(8) \] \[ S = 192 + 192 + 128 \] \[ S = 512 \text{ ft}^2 \]
Step 2: Surface area of the right triangular prism
The surface area of the right triangular prism is determined by finding the area of the two triangular bases and the three rectangular faces.
Area of the triangular base:
The base of the triangular prism is a right triangle. The width of the prism is given as 8 feet, and the total height of the composite figure suggests that the height of the triangular prism is \(20 - 8 = 12\) feet.
Using the Pythagorean theorem:
\[ \text{Height of triangle} = 12 \text{ ft (it matches the total height of the prism)} \] \[ \text{Base of triangle} = 8 \text{ ft} \]
The area \(A\) of a right triangle is:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Thus,
\[ A = \frac{1}{2} \times 8 \times 12 = 48 \text{ ft}^2 \]
Since there are two triangular bases:
\[ \text{Total area of the triangular bases} = 2 \times 48 = 96 \text{ ft}^2 \]
Area of the rectangular faces:
The remaining three faces are rectangles:
-
Two faces with area equal to the height of the prism multiplied by the length (the slanted side):
- The slanted side (hypotenuse) of the triangular base is given as 15 feet.
- The areas of the two rectangles joining both bases:
- Each area = \(15 \times 12 = 180 \text{ ft}^2\)
- Two such faces:
\[ 2 \times 180 = 360 \text{ ft}^2 \]
-
One rectangular face with area equal to the base width times the height:
- Area = \(8 \times 12 = 96 \text{ ft}^2\)
Total surface area of the triangular prism:
Combining them:
\[ \text{Total area of rectangular faces} = 360 + 96 = 456 \text{ ft}^2 \] \[ S_{\text{triangular prism}} = 96 + 456 = 552 \text{ ft}^2 \]
Step 3: Combine both surface areas
Adding both surface areas together:
\[ S_{\text{total}} = S_{\text{rectangular prism}} + S_{\text{triangular prism}} \] \[ S_{\text{total}} = 512 + 552 = 1064 \text{ ft}^2 \]
Thus, the surface area of the composite figure is
\[ \boxed{1064} \text{ ft}^2 \]