To find the total surface area of the composite figure, we need to calculate the surface area of the larger rectangular prism and then adjust for the missing section.
Step 1: Surface Area of the Larger 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
For the larger rectangular prism:
- \( l = 15 \) inches
- \( w = 6 \) inches
- \( h = 12 \) inches
Calculating the surface area:
\[ S = 2(15)(6) + 2(15)(12) + 2(6)(12) \]
Calculating each term:
- \( 2(15)(6) = 180 \)
- \( 2(15)(12) = 360 \)
- \( 2(6)(12) = 144 \)
Adding these, we get:
\[ S = 180 + 360 + 144 = 684 \text{ square inches} \]
Step 2: Surface Area of the Missing Section
The missing section also is a rectangular prism. We will calculate the area of the visible surface area that is now exposed because this part is missing.
The dimensions of the missing rectangular prism:
- Length = 7 inches
- Width = 6 inches (same as larger prism)
- Height = 6 inches
The surface area of the missing section would be:
\[ S_{missing} = 2lw + 2lh + 2wh \] Here, since we are going to be adding the surfaces that are exposed due to the absence of the missing section, we need to note that the bottom face of the missing section will not contribute exposed surface area since it is in contact with the larger prism.
So we will calculate:
\[ S_{missing, exposed} = 2lw + 2lh \]
Calculating this: \[ S_{missing} = 2(7)(6) + 2(7)(6) \]
Calculating, we find:
- \( 2(7)(6) = 84 \)
- \( 2(7)(6) = 84 \)
Thus: \[ S_{missing, exposed} = 84 + 84 = 168 \text{ square inches} \]
Step 3: Total Surface Area of the Composite Figure
Finally, to find the total surface area of the composite figure, we will sum the surface areas of the larger prism and the exposed surface area of the missing section:
\[ \text{Total Surface Area} = S + S_{missing, exposed} \]
Thus: \[ \text{Total Surface Area} = 684 + 168 = 852 \text{ square inches} \]
Therefore, the total surface area of the figure is:
\[ \boxed{852} \text{ square inches} \]