SOLUTION: designing a pool where the area 16 and the perimeter is 20.

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Question 413587: designing a pool where the area 16 and the perimeter is 20.
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
designing a pool where the area 16 and the perimeter is 20.
.
assuming it's a rectangular pool...
.
Let W = width
and L = length
.
from knowledge of perimeter:
2(W+L) = 20
W+L = 10 (equation 1)
.
from knowledge of area:
WL = 16 (equation 2)
.
Solving equation 1 for L we get:
L = 10-W
Substitute into equation 2:
W(10-W) = 16
W^2+10W = 16
W^2+10W-16 = 0
Applying the quadratic formula we get:
W = {1.4, -11.4}
.
Find L by substitutiting above into equation 1:
1.4+L = 10
L = 8.6
.
Dimensions are: 1.4 by 8.6
.
Details of quadratic follows
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aW%5E2%2BbW%2Bc=0 (in our case 1W%5E2%2B10W%2B-16+=+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=%2810%29%5E2-4%2A1%2A-16=164.

Discriminant d=164 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-10%2B-sqrt%28+164+%29%29%2F2%5Ca.

W%5B1%5D+=+%28-%2810%29%2Bsqrt%28+164+%29%29%2F2%5C1+=+1.40312423743285
W%5B2%5D+=+%28-%2810%29-sqrt%28+164+%29%29%2F2%5C1+=+-11.4031242374328

Quadratic expression 1W%5E2%2B10W%2B-16 can be factored:
1W%5E2%2B10W%2B-16+=+1%28W-1.40312423743285%29%2A%28W--11.4031242374328%29
Again, the answer is: 1.40312423743285, -11.4031242374328. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B10%2Ax%2B-16+%29