An illustration shows the unfolded version of a rectangular prism composed of 6 rectangles. There are four adjoining rectangles positioned vertically. The first and third are similar and bigger. The second and fourth are similar and smaller. The third rectangle is labeled as 6 feet on the left side. The fourth rectangle is labeled as 8 feet on the bottom side. The fourth rectangle shares the left and right sides with two similar rectangles, one on each side. The rectangle on the left is labeled as 3.5 feet on the left side.
What is the surface area of this prism?
(1 point)
Responses
194 ft.2
194 ft. squared
168 ft.2
, 168 ft. squared
97 ft.2
, 97 ft. squared
35 ft.2
, 35 ft. squared
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The surface area of a rectangular prism can be calculated by finding the area of each of its six faces and adding them together.
From the given labels, we can see that the first and third rectangles (positioned vertically) are similar and bigger, and the second and fourth rectangles are similar and smaller. Let's label the sides of the rectangles as follows:
- First rectangle: 6 feet (left) by x feet (bottom)
- Second rectangle: 3.5 feet (left) by y feet (bottom)
- Third rectangle: 6 feet (left) by x feet (top)
- Fourth rectangle: y feet (left) by 8 feet (bottom)
We can use the fact that the second and fourth rectangles are similar to find y:
y/3.5 = x/8
y = 1.6x
Substituting this into the labels for the second and fourth rectangles, we get:
- Second rectangle: 3.5 feet (left) by 1.6x feet (bottom)
- Fourth rectangle: 1.6x feet (left) by 8 feet (bottom)
Now we can calculate the area of each face:
- Top face (third rectangle): 6x square feet
- Bottom face (fourth rectangle): 1.6x * 8 = 12.8x square feet
- Left face (first, second, and fourth rectangles): 6y + 3.5y + 1.6x * 8 = 22.8y square feet
- Right face (same as left face): 22.8y square feet
- Front face (first and second rectangles): 6 * x + 3.5 * y = 6x + 5.6x = 11.6x square feet
- Back face (same as front face): 11.6x square feet
Adding up the areas, we get:
6x + 12.8x + 22.8y + 22.8y + 11.6x + 11.6x = 30.6x + 45.6y
Substituting y = 1.6x, we get:
30.6x + 45.6(1.6x) = 110.4x
Therefore, the surface area of the rectangular prism is 110.4 square feet.
Answer: 110.4 ft.2 or 110.4 ft. squared