To calculate the total surface area of the composite figure, we need to consider the surface area of the larger rectangular prism and then subtract the area that is no longer exposed due to the removal of the smaller rectangular prism.
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 length
- \( w \) is width
- \( h \) is height
For the larger prism:
- Length \( l = 15 \) inches
- Width \( w = 6 \) inches
- Height \( h = 12 \) inches
Plugging in these values:
\[ SA_{large} = 2(15)(6) + 2(15)(12) + 2(6)(12) \]
Calculating each term:
- \( 2(15)(6) = 180 \)
- \( 2(15)(12) = 360 \)
- \( 2(6)(12) = 144 \)
Now, adding them together:
\[ SA_{large} = 180 + 360 + 144 = 684 \text{ in}^2 \]
Step 2: Calculate the surface area of the smaller rectangular prism that was removed.
For the missing section (smaller prism):
- Length \( l = 7 \) inches
- Width \( w = 6 \) inches (same as the larger prism)
- Height \( h = 6 \) inches
Using the surface area formula again:
\[ SA_{small} = 2(7)(6) + 2(7)(6) + 2(6)(6) \]
Calculating each term:
- \( 2(7)(6) = 84 \)
- \( 2(7)(6) = 84 \) (same calculation as above)
- \( 2(6)(6) = 72 \)
Now, adding them together:
\[ SA_{small} = 84 + 84 + 72 = 240 \text{ in}^2 \]
Step 3: Calculate the areas that are no longer exposed due to the removal of the smaller prism.
The area of the top face of the smaller prism, which is no longer exposed, is:
\[ Area_{top} = l \times w = 7 \times 6 = 42 \text{ in}^2 \]
The area of the four vertical faces is now counted differently as one section of the prism is removed. The smaller prism has four vertical sides:
- Two sides with dimensions (height \(h\) = 6, width of smaller prism \(w = 6\)):
- Area of both = \( 2 \times (6 \times 6) = 72 \text{ in}^2 \)
- Two sides with dimensions (height \(h\) = 6, length of smaller section \(l = 7\)):
- Area of both = \( 2 \times (6 \times 7) = 84 \text{ in}^2 \)
Combining these areas of exposed faces:
\[ Total_{removed} = Area_{top} + 2 \cdot (Height \cdot Width) + 2 \cdot (Height \cdot Length) \]
\[ Total_{removed} = 42 + 72 + 84 = 198 \text{ in}^2 \]
However, this count up includes the bottom face. We need total removed in terms of visible surface area.
The top face of the smaller section which is subtracted must be added to the overall surface area calculation back so the exposed region must be recalculated.
Step 4: Final Calculation of Total Surface Area
The total surface area of the figure can now be found using:
\[ Total\ SA = SA_{large} - Area_{top} + Exposed_Area = SA_{large} + Surface_Area = (684 - 42 + 72 + 84) = 684 + 114 = 756 inch^2 \]
Thus, the total surface area of the modified figure is:
\[ \boxed{756 \text{ in}^2} \]