SOLUTION: A rectangle has a length of 31 feet less than 6 times its width. If the area of the rectangle is 4470 square feet, find the length of the rectangle.

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Question 952637: A rectangle has a length of 31 feet less than 6 times its width. If the area of the rectangle is 4470 square feet, find the length of the rectangle.
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
W=width; L=length=6W-31ft;
A=L*W
4470sqft=(6W-31ft)(W)
4470ft%5E2=6W%5E2-31W
0=6W%5E2-31W-4470ft%5E2
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aW%5E2%2BbW%2Bc=0 (in our case 6W%5E2%2B-31W%2B-4470+=+0) has the following solutons:

W%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=%28-31%29%5E2-4%2A6%2A-4470=108241.

Discriminant d=108241 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--31%2B-sqrt%28+108241+%29%29%2F2%5Ca.

W%5B1%5D+=+%28-%28-31%29%2Bsqrt%28+108241+%29%29%2F2%5C6+=+30
W%5B2%5D+=+%28-%28-31%29-sqrt%28+108241+%29%29%2F2%5C6+=+-24.8333333333333

Quadratic expression 6W%5E2%2B-31W%2B-4470 can be factored:
6W%5E2%2B-31W%2B-4470+=+6%28W-30%29%2A%28W--24.8333333333333%29
Again, the answer is: 30, -24.8333333333333. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+6%2Ax%5E2%2B-31%2Ax%2B-4470+%29

The width is 30 feet.
L=6W-31ft+6(30ft)-31ft=180ft-31ft=149ft ANSWER: The length is 149 feet
CHECK:
A=L*W
4470sqft=149ft*30ft
4470sqft=4470sqft