SOLUTION: A 1200 square foot rectangular garden is enclosed with 150 ft of fencing. Find the dimensions of the garden to two decimal places.

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Question 762657: A 1200 square foot rectangular garden is enclosed with 150 ft of fencing. Find the dimensions of the garden to two decimal places.
Answer by Stitch(470) About Me  (Show Source):
You can put this solution on YOUR website!
The equation for the area of a rectangle is A=L*W
The equation for the perimeter of a rectangle is P=2L+2W
Given: A=1200 & P=150
Equation 1: 1200+=+L%2AW%29%29%29%0D%0AEquation+2%3A+%7B%7B%7B150+=+2L+%2B+2W
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Start by solving equation 2 for one of the variables.
Equation 2: 150+=+2L+%2B+2W
Subtract 2L from both sides.
150+-+2L+=+2W
Factor out a 2 on the left side of the equation.
2%2A%2875-L%29+=+2W
Divide both sides by 2.
75+-+L+=+W
Now plug (75-L) into equation 1 for W.
Equation 1: 1200+=+L%2AW%29%29%29%0D%0A%7B%7B%7B1200+=+L%2A%2875+-+L%29
Multiply the L through.
1200+=+75L+-+L%5E2
Subtract 1200 from both sides.
0+=+-L%5E2+%2B+75L+-+1200
Now you can use the quadratic equation to solve for L.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aL%5E2%2BbL%2Bc=0 (in our case -1L%5E2%2B75L%2B-1200+=+0) has the following solutons:

L%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2875%29%5E2-4%2A-1%2A-1200=825.

Discriminant d=825 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-75%2B-sqrt%28+825+%29%29%2F2%5Ca.

L%5B1%5D+=+%28-%2875%29%2Bsqrt%28+825+%29%29%2F2%5C-1+=+23.1385933836549
L%5B2%5D+=+%28-%2875%29-sqrt%28+825+%29%29%2F2%5C-1+=+51.8614066163451

Quadratic expression -1L%5E2%2B75L%2B-1200 can be factored:
-1L%5E2%2B75L%2B-1200+=+-1%28L-23.1385933836549%29%2A%28L-51.8614066163451%29
Again, the answer is: 23.1385933836549, 51.8614066163451. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B75%2Ax%2B-1200+%29

You will end up with two values for L.
L is equal to 23.14 & 51.86
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Using the values for L, solve for W.
75+-+L+=+W
75+-+23.14+=+W
highlight%2851.86+=+W%29
or
75+-+51.86+=+W
highlight_green%2823.14+=+W%29
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In other words the length and width can be switched but the dimensions have to be 23.14ft and 51.86ft