To find the total surface area of the composite figure, we need to calculate the surface areas of the large rectangular prism and the missing section, then adjust for the areas that are not exposed due to the removal of the missing section.
Step 1: Surface Area of the Large Rectangular Prism
The surface area \( SA \) of a rectangular prism is given by the formula:
\[ SA = 2(lw + lh + wh) \]
where:
- \( l \) is the length,
- \( w \) is the width,
- \( h \) is the height.
For the large rectangular prism:
- Length \( l = 15 \) in,
- Width \( w = 6 \) in,
- Height \( h = 12 \) in.
Calculating the surface area:
\[ SA = 2(15 \cdot 6 + 15 \cdot 12 + 6 \cdot 12) \]
Calculating each term:
- \( 15 \cdot 6 = 90 \)
- \( 15 \cdot 12 = 180 \)
- \( 6 \cdot 12 = 72 \)
Summing those up:
\[ SA = 2(90 + 180 + 72) \] \[ SA = 2(342) = 684 \text{ in}^2 \]
Step 2: Surface Area of the Missing Section
The missing section is also a rectangular prism with:
- Length \( l = 7 \) in,
- Width \( w = 6 \) in,
- Height \( h = 6 \) in.
Calculating the surface area:
\[ SA_{\text{missing}} = 2(7 \cdot 6 + 7 \cdot 6 + 6 \cdot 6) \]
Calculating each term:
- \( 7 \cdot 6 = 42 \)
- \( 6 \cdot 6 = 36 \)
Thus,
\[ SA_{\text{missing}} = 2(42 + 42 + 36) = 2(120) = 240 \text{ in}^2 \]
Step 3: Correcting for the Area Removed
Since this missing section is removed from the top of the large prism, we need to consider the area of the base of the missing section that is now exposed:
- The face of the missing section that was removed has an area of \( l \times w = 7 \times 6 = 42 \text{ in}^2 \).
Step 4: Total Surface Area Calculation
The total surface area of the composite figure is the surface area of the large prism plus the exposed areas minus the area of the missing section that was removed:
\[ SA_{\text{total}} = SA + SA_{\text{missing}} - \text{Area}{\text{removed}} \] \[ SA{\text{total}} = 684 + 240 - 42 \] \[ SA_{\text{total}} = 882 \text{ in}^2 \]
Thus, the total surface area of the figure is \( \boxed{882} \text{ in}^2 \).