SOLUTION: A rectangular piece of sheet metal with an area of 1200 in^2 is to be bent into a cylindrical length of stovepipe having a volume of 600 in^3. What are the dimensions of the sheet

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Question 66751This question is from textbook College Algebra
: A rectangular piece of sheet metal with an area of 1200 in^2 is to be bent into a cylindrical length of stovepipe having a volume of 600 in^3. What are the dimensions of the sheet metal?
I am to come up with a system of equations to solve this problem. The sides of the rectangle are labeled x and y.
I know that {x*y=1200 in^2}; x would be the circumference of the cylinder which would mean that {x=2(PI)r}; the volume would be {(PI)r^2(y)=600 in^3}. I just can't figure out how to show it all in terms of y and x to figure out the equation. Thanks.
This question is from textbook College Algebra

Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
You are correct in what you have done so far.
Let's start with the volume of the cylindical pipe.
%28pi%29r%5E2y+=+600 Substitute r here with: r+=+x%2F2%28pi%29 from x+=+2%28pi%29r
%28pi%29%28x%2F2%28pi%29%29%5E2y+=+600 Simplify.
%28x%5E2%2F4%28pi%29%29y+=+600 Now substitute the y here with y+=+1200%2Fx from the area, xy+=+1200
%28x%5E2%2F4%28pi%29%29%281200%2Fx%29+=+600 Simplify.
1200x%2F4%28pi%29+=+600 Multiply both sides by 4%28pi%29
1200x+=+2400%28pi%29 Divide both sides by 1200.
x+=+2%28pi%29 and, from above:
y+=+1200%2Fx
y+=+1200%2F2%28pi%29
y+=+600%2F%28pi%29
The dimensions of the rectangular metal sheet are:
2%28pi%29 by 600%2F%28pi%29 inches. or...using %28pi%29+=+3.14 as an approximation:
2%283.14%29 by 600%2F3.14 or 6.28 by 191.08 inches.