SOLUTION: Find a polynomial for the surface area of the right rectangular solid. length - 7 width - a height - 4 it looks to me like the right rectangular solid is the length

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Question 166481This question is from textbook Introductory Algebra
: Find a polynomial for the surface area of the right rectangular solid.
length - 7
width - a
height - 4
it looks to me like the right rectangular solid is the length This question is from textbook Introductory Algebra

Found 2 solutions by Earlsdon, gonzo:
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
The total surface area of a right rectangular solid (aka prism) consists of the sum of the areas of the six faces of the solid.
These six faces are composed of three congruent pairs of faces:
The top & bottom, The back & front, and the two sides.
So you can find the are of one each of these then multiply by two for the total surface area.
Area of the top:
A%5Bt%5D+=+L%2AW Substitute L = 7 and W = a.
A%5Bt%5D+=+7a Now multiply by 2 to include the bottom.
A%5Bt%5D+=+2%287a%29
A%5Bt%5D+=+14a
Area of the front:
A%5Bf%5D+=+W%2Ah Substitute: W = a and h = 4.
A%5Bf%5D+=+a%2A4 or A%5Bf%5D+=+4a Now multiply by 2 to include the back.
A%5Bf%5D+=+2%284a%29
A%5Bf%5D+=+8a
Area of the side:
A%5Bs%5D+=+L%2Ah Substitute L = 7 and h = 4.
A%5Bs%5D+=+7%2A4 Multiply by 2 to include the other side.
A%5Bs%5D+=+2%2828%29
A%5Bs%5D+=+56
Now add the three areas computed above to get the total surface area:
A%5BT%5D+=+14a%2B8a%2B56 Combine like-terms.
A%5BT%5D+=+22a%2B56

Answer by gonzo(654) About Me  (Show Source):
You can put this solution on YOUR website!
surface area (S) of a right rectangular solid is:
S = 2*L*W + 2*L*H + 2*W*H (equation 1)
where:
L = length
W = width
H = height
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you have:
L = 7
W = a
H = 4
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formula becomes:
S = 2*7*a + 2*7*4 + 2*a*4
simplify:
S = 14*a + 56 + 8*a
combine like terms:
S = 22*a + 56
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answer is:
S = 22*a + 56 (equation 2)
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to prove, substitute any value for a in equation 2:
let a = 15
S = 22*a + 56 (equation 2)
S = 22*15 + 56
S = 330 + 56
S = 386
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since a = 15, then W = 15 because W = a
go back to equation 1 and substitute 15 for W:
S = 2*L*W + 2*L*H + 2*W*H (equation 1)
where:
L = length
W = width
H = height
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you have:
L = 7
W = 15
H = 4
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equation 1 becomes:
S = 2*7*15 + 2*7*4 + 2*15*4
perform indicated operations:
S = 210 + 56 + 120
S = 386
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surface area is the same using both equations so:
equations are good.
formula for surface area of right rectangle in terms of a is:
S = 22*a + 56
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