To find the total surface area of the composite figure, we need to calculate the surface area of the larger rectangular prism, subtract the surface area of the missing section, and then add the area of the new exposed faces created by the removal of the section.
Step 1: Surface Area of the Larger Rectangular Prism
The formula for the surface area \( S \) of a rectangular prism is:
\[ S = 2(lw + lh + wh) \]
Where:
- \( l \) is the length
- \( w \) is the width
- \( h \) is the height
For the larger rectangular prism:
- Length \( l = 15 \) inches
- Width \( w = 6 \) inches
- Height \( h = 12 \) inches
Calculating the surface area:
\[ S = 2(15 \times 6 + 15 \times 12 + 6 \times 12) \] \[ = 2(90 + 180 + 72) \] \[ = 2(342) \] \[ = 684 \text{ in}^2 \]
Step 2: Surface Area of the Missing Section
For the missing section (also a rectangular prism):
- Length \( l = 7 \) inches
- Width \( w = 6 \) inches
- Height \( h = 6 \) inches
Calculating the surface area of the missing section:
\[ S = 2(7 \times 6 + 7 \times 6 + 6 \times 6) \] \[ = 2(42 + 42 + 36) \] \[ = 2(120) \] \[ = 240 \text{ in}^2 \]
Step 3: Accounting for New Exposed Faces
When the section is removed, we have new exposed faces:
- The bottom face of the missing section (which is \( 7 \times 6 = 42 \) square inches).
- The front face of the missing section (which is \( 7 \times 6 = 42 \) square inches).
- The back face of the missing section (which is also \( 7 \times 6 = 42 \) square inches).
Total area of the new exposed faces:
\[ 42 + 42 + 42 = 126 \text{ in}^2 \]
Step 4: Final Calculation of Total Surface Area
Now, combining all these together:
\[ \text{Total Surface Area} = \text{Surface Area of Larger Prism} - \text{Surface Area of Missing Section} + \text{Area of New Exposed Faces} \] \[ = 684 - 240 + 126 \] \[ = 684 - 240 + 126 = 570 \text{ in}^2 \]
Thus, the total surface area of the composite figure is:
\[ \boxed{570} \text{ in}^2 \]