The figure below is a net for a right rectangular prism. Its surface area is 280 cm² and the area of some of the faces are filled in below. Find the area of the missing faces, and the missing dimension.On the tactile printout of the following graphs, d is 50 centimeters squared, c is 30 centimeters squared, g is 5 centimeters, and f is 6 centimeters. The missing face is labeled A and the missing dimension is labeled with a question mark.
To find the missing face labeled A, we need to calculate the total surface area of the right rectangular prism and then subtract the known areas of the faces from it.
Given:
Total surface area = 280 cm²
Face d = 50 cm²
Face c = 30 cm²
Face g = 5 cm²
Face f = 6 cm²
Let the missing face A be x cm²
Total surface area = 2(Length x Width) + 2(Width x Height) + 2(Length x Height)
280 = 2(30)+2(5h)+2(6l)
280 = 60 + 10h + 12l
220 = 10h + 6l
g=5 not included
f=6 not included
d= 50 not included
c=30 not included
From the given values, we know:
g = 5 cm
f = 6 cm
d = 50 cm²
c = 30 cm²
Let the missing dimension be represented as l (length) and w (width). We need to find the values of l and w.
From the formula for total surface area of a right rectangular prism, we have:
2lw + 2lh + 2wh = 280
Plugging in the values of the known faces:
2lw + 2lh + 2wh = 280
2lw + 2(5)w + 2(6)l + 2(30) = 280
Simplify:
2lw + 10w + 12l + 60 = 280
2lw + 10w + 12l = 220
Using the given dimensions of the known faces, we have:
d = lw = 50
c = lh = 30
Using the above relationships, we can substitute back using the values of c and d:
l = 30/w
w = 50/l
Substitute these relationships back into the equation:
2(30/w)w + 10w + 12(30/w) = 220
60 + 10w + 360/w = 220
Rearrange and simplify:
10w + 360/w = 160
10w^2 - 160w + 360 = 0
w^2 - 16w + 36 = 0
(w - 6)(w - 6) = 0
w = 6
Therefore, the missing dimension w (width) is 6 cm. To find the missing length l, we can use the relationship derived earlier:
l = 30/6 = 5 cm
Thus, the missing dimension l (length) is 5 cm.
Now we can calculate the missing face A:
A = lw
A = 5 cm * 6 cm
A = 30 cm²
Therefore, the missing face A has an area of 30 cm².
what about the length of each missing edge?
To find the length of the missing edge of the right rectangular prism, we can use the dimensions we have already calculated for the width (w) and length (l) of the prism.
The missing dimension labeled with a question mark corresponds to the height of the prism.
Given:
Width (w) = 6 cm
Length (l) = 5 cm
Let's label the missing height dimension as h.
Using the formula for the total surface area of a right rectangular prism:
2lw + 2lh + 2wh = 280
Substitute in the known values for w, l, and the missing dimension h:
2(5)(6) + 2(5)h + 2(6)h = 280
60 + 10h + 12h = 280
22h = 220
h = 220 / 22
h = 10
Therefore, the missing dimension 'h' or the height of the right rectangular prism is 10 cm.
In summary, the missing edge lengths of the right rectangular prism are:
- Length (l) = 5 cm
- Width (w) = 6 cm
- Height (h) = 10 cm