The figure below is a net for a right rectangular prism. Its surface area is 246 ft² and the area of some of the faces are filled in below. Find the area of the missing faces, and the missing dimension.
72 ft²
72 ft²
27 ft²
27 ft²
9 ft
9 ft
8 ft
8 ft
8 ft
8 ft
?
?
A
A
The area of each missing face is
ft²
The length of each missing edge is
ft
To find the area of the missing faces, we need to subtract the area of the given faces from the total surface area.
Total surface area = 246 ft²
Given area = 2(72 ft²) + 2(27 ft²) + 2(8 ft x 9 ft) = 144 ft² + 54 ft² + 144 ft² = 342 ft²
Missing area = 246 ft² - 342 ft² = -96 ft²
This means that there is an error in the given values, as we cannot have negative area. We can try to find the missing dimension to see if there is a mistake.
Let the dimensions of the prism be length (L), width (W), and height (H).
From the net, we can see that:
- L = 2(8 ft) + 9 ft = 25 ft
- WH = 27 ft², so either W or H is 9 ft and the other is 3 ft.
- LW = 72 ft², so either L or W is 8 ft and the other is 9 ft.
We can check these possibilities using the equation for total surface area:
Total surface area = 2(LW + LH + WH)
If L = 25 ft, W = 9 ft, and H = 3 ft, then:
Total surface area = 2(25 ft x 9 ft + 25 ft x 3 ft + 9 ft x 3 ft) = 246 ft²
This matches the given value, so this is the correct solution.
The missing dimension is the width, which is 8 ft. Therefore, the missing faces have areas of 9 ft² and 25 ft².
Missing area = 9 ft² + 25 ft² = 34 ft²
Missing edge length = 8 ft
Final answer:
The area of each missing face is 9 ft² and 25 ft².
The length of each missing edge is 8 ft.
To find the area of the missing faces, we need to subtract the area of the filled-in faces from the total surface area of the prism.
Total surface area of the prism = 246 ft²
Area of the filled-in faces = 72 ft² + 72 ft² + 27 ft² + 27 ft² + 9 ft + 9 ft = 216 ft²
To find the area of the missing faces, we subtract the area of the filled-in faces from the total surface area:
Area of the missing faces = Total surface area - Area of filled-in faces
= 246 ft² - 216 ft²
= 30 ft²
The area of each missing face is 30 ft².
To find the length of each missing edge, we need to determine the dimensions of the net. Based on the given information, we can see that the missing faces are labeled with "A." This suggests that the dimensions of those faces are the same.
Since the prism is a right rectangular prism, let's assume the missing dimension is the height (h), which is the same for both missing faces.
We can use the formula for the surface area of a rectangular prism to find the height:
Surface area = 2lw + 2lh + 2wh
Given information:
l = 8 ft
w = 9 ft
Surface area = 246 ft²
246 ft² = 2(8 ft)(9 ft) + 2(8 ft)(h) + 2(9 ft)(h)
= 144 ft² + 16h + 18h
Combine like terms:
246 ft² = 144 ft² + 34h
Simplify:
102 ft² = 34h
Divide both sides by 34:
3 ft = h
The length of each missing edge (height) is 3 ft.