1. The surface area of the figure is given by the total area of the exterior surfaces of the composite solid - here, we have a triangular prism atop a rectangular prism.
2. Let's start with the surface area of the rectangular prism. There are five rectangles (front and back; left and right; bottom). Note that we do not count the top rectangle in our calculations as it is covered by the bottom of the triangular prism and thus not included in the surface area (as it is not visible from the outside of the shape).
The general formula for the area of a rectangle is:
Area = lw, where l is the length and w is the width.
a) Front and back rectangles.
Here, l = 10 and w = 10. Therefor:
Area = 10*10 = 100 ft^2
Since there are two rectangles, we should multiply this value by 2 to get:
Area of front and back rectangles
= 100*2
= 200 ft^2
b) Left and right rectangles
Here, l = 14 and w = 10. Therefor:
Area = 14*10 = 140 ft^2
Again, since there are two such rectangles, we need to multiply our answer by 2:
Area of left and right rectangles
= 140*2
= 280 ft^2
c) Bottom rectangle
Here, l = 14 and w = 10. Therefor:
Area of bottom rectangle
= 14*10
= 140 ft^2
(Remember, we do not need to include the area of the top rectangle in the total surface area, thus we do not need to multiply our answer by 2 as we had done with a) and b). )
Thus, we can add the areas found in a), b) and c) to get:
Surface area of rectangular prism
= 200 + 280 + 140
= 620 ft^2
3. Now, let's calculate the surface area of the triangular prism. This may be broken up into two parts: front and back triangles, and left and right rectangles (we do not count the bottom rectangle as it sits atop the top rectangle of the rectangular prism).
i) Front and back triangles
The formula for the area of a triangle is:
Area = (1/2)bh, where b is the length of the base and h is the height.
In our case, the base is 10 ft and the height is 8 ft. Thus:
Area = (1/2)*10*8
= 5*8
= 40 ft^2
Since we have two such triangles, we should multiply this value by 2:
Area = 40*2 = 80 ft^2
ii) Left and right rectangles
Area of rectangle = lw
Here, the length is 14 ft and the width is 9 ft. Thus:
Area = 14*9 = 126 ft^2
Since we have two such rectangles, we shall multiply this value by 2 to get:
Area = 126*2 = 252 ft^2
Now, we can add i) and ii) together to get:
Surface area of triangular prism = 80 + 252 = 332 ft^2
4. Now all we have to do is add the surface area of the triangular prism to that of the rectangular prism to get the total surface area:
Total surface area
= 620 + 332
= 952 ft^2
Therefor, the surface area of the composite figure is 952 ft^2.