To find the total surface area of the composite 3D figure, we need to calculate the surface area of the larger rectangular prism and then subtract the area of the top face of the missing rectangular prism, while accounting for any additional exposed area where the missing section creates new surfaces.
Step 1: Calculate the surface area of the larger rectangular prism.
The formula for the surface area \(SA\) of a rectangular prism is: \[ SA = 2lw + 2lh + 2wh \] where \(l\) is the length, \(w\) is the width, and \(h\) is the height.
For the larger rectangular prism:
- Length \(l = 15\) inches
- Width \(w = 6\) inches
- Height \(h = 12\) inches
\[ SA_{larger} = 2(15 \times 6) + 2(15 \times 12) + 2(6 \times 12) \]
Calculating each term:
- \(2(15 \times 6) = 2(90) = 180\)
- \(2(15 \times 12) = 2(180) = 360\)
- \(2(6 \times 12) = 2(72) = 144\)
Adding them up: \[ SA_{larger} = 180 + 360 + 144 = 684 \text{ in}^2 \]
Step 2: Calculate the surface area of the missing rectangular prism.
For the missing section:
- Length \(l = 7\) inches
- Width \(w = 6\) inches
- Height \(h = 6\) inches
Calculating the surface area of the missing section: \[ SA_{missing} = 2lw + 2lh + 2wh = 2(7 \times 6) + 2(7 \times 6) + 2(6 \times 6) \]
Calculating each term:
- \(2(7 \times 6) = 2(42) = 84\) (twice, since there are two of these faces)
- \(2(6 \times 6) = 2(36) = 72\)
Adding them up: \[ SA_{missing} = 84 + 84 + 72 = 240 \text{ in}^2 \]
Step 3: Calculate the total surface area of the composite figure.
Now, the total surface area of the composite figure (\(SA_{total}\)) will be the surface area of the larger prism minus the area of the top face of the missing section plus the area of the newly exposed surfaces created by the removal (the sides of the missing section).
The top face of the missing section (which is also on the top of the larger prism), has an area: \[ \text{Area of top face} = l \times w = 7 \times 6 = 42 \text{ in}^2 \]
Now, we add the exposed side areas from the missing section, which contribute to the surface area:
- Each of the two sides has an area of \(6 \times 6 = 36\) in\(^2\) and there are two sides.
- The front and back faces (which were not part of the surface of the original prism) have an area of \(7 \times 6 = 42\) in\(^2\) each, and there are two of these faces.
Total extra area from the missing section: \[ \text{Total exposed area from missing section} = 2(36) + 2(42) = 72 + 84 = 156 \text{ in}^2 \]
Thus, the total surface area of the composite figure is: \[ SA_{total} = SA_{larger} - \text{Area of top face} + \text{Total exposed area from missing section} \] \[ SA_{total} = 684 - 42 + 156 = 798 \text{ in}^2 \]
Final Answer:
The total surface area of the composite figure is \( \boxed{798} \) in\(^2\).