Question - Calculating Surface Area of Complex 3D Shape
Solution:
The image displays a 3-dimensional shape composed of two rectangular prisms. To find the surface area of the entire object, we must calculate the area of every face that is visible or implied (hidden).First, let's label the dimensions of each part:- The larger prism has a width of 3 meters, a depth of 3 meters, and a height of 7 meters.- The smaller prism is attached to the larger one; it has a width of 1 meter, the same depth as the larger prism (3 meters), and a height of 3 meters.Now let's calculate the surface area of each face:For the larger prism:1. Front and back faces: $$2 \times (width \times height) = 2 \times (3m \times 7m) = 2 \times 21m^2 = 42m^2$$2. Left and right faces: These will be the sides without the smaller prism attached. Only one full side is present because the other one is partially covered by the smaller prism. $$1 \times (depth \times height) = 1 \times (3m \times 7m) = 1 \times 21m^2 = 21m^2$$3. Top and bottom faces: Since it's a prism, the top and bottom faces are the same size. $$2 \times (width \times depth) = 2 \times (3m \times 3m) = 2 \times 9m^2 = 18m^2$$For the smaller prism:1. Front and back faces: Note that the front face is visible, but the back face is against the larger prism and isn't seen. $$1 \times (width \times height) = 1 \times (1m \times 3m) = 1 \times 3m^2 = 3m^2$$2. Left and right faces: Both faces are visible. $$2 \times (depth \times height) = 2 \times (3m \times 3m) = 2 \times 9m^2 = 18m^2$$3. Top face: The bottom face is on top of the larger prism, so only the top face counts. $$1 \times (width \times depth) = 1 \times (1m \times 3m) = 1 \times 3m^2 = 3m^2$$Lastly, we must subtract the area of the face that's shared between both prisms, as we have accounted for it twice (once in the larger prism's surface area, and once in the smaller prism's surface area). This is the face on the side where the smaller prism touches the larger one:Shared face area: $$width \times height = 1m \times 3m = 3m^2$$Adding everything together:$$SA = (42m^2 + 21m^2 + 18m^2) + (3m^2 + 18m^2 + 3m^2) - 3m^2 = 102m^2$$So the total surface area of the object is $$102 \text{ square meters}$$.
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