SOLUTION: Please help. Thomas is going to make an open-top box by cutting equal squares from the four corners of an 11 inch by 14 inch sheet of cardboard and folding up the sides. If the ar

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Question 178715: Please help.
Thomas is going to make an open-top box by cutting equal squares from the four corners of an 11 inch by 14 inch sheet of cardboard and folding up the sides. If the area of the base is to be 80 square inches, then what size square should be cut from each corner?
Found 2 solutions by nerdybill, ankor@dixie-net.com:
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
Thomas is going to make an open-top box by cutting equal squares from the four corners of an 11 inch by 14 inch sheet of cardboard and folding up the sides. If the area of the base is to be 80 square inches, then what size square should be cut from each corner?
.
It's important to draw a diagram of the problem.
Once you do that, it'll be easier to "see" the solution.
.
Let x = measure of one side of ths square that is being cut from each corner
then area of the base would be
(11-2x)(14-2x) = 80
FOIL the left:
154 - 22x - 28x + 4x^2 = 80
154 - 50x + 4x^2 = 80
Rearranging...
4x^2 - 50x + 154 = 80
4x^2 - 50x + 74 = 0
2x^2 - 25x + 37 = 0
.
Since we can't factor you must apply the quadratic equation. Doing so yields:
x = {10.785, 1.715}
As you can see, there are two possible solutions -- but, the one that makes the most sense would be 1.715 inches the other will make an awfully funny shape box.
.
Below are the details of the quadratic:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B-25x%2B37+=+0) has the following solutons:

x%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=%28-25%29%5E2-4%2A2%2A37=329.

Discriminant d=329 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--25%2B-sqrt%28+329+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-25%29%2Bsqrt%28+329+%29%29%2F2%5C2+=+10.7845892868043
x%5B2%5D+=+%28-%28-25%29-sqrt%28+329+%29%29%2F2%5C2+=+1.71541071319574

Quadratic expression 2x%5E2%2B-25x%2B37 can be factored:
2x%5E2%2B-25x%2B37+=+2%28x-10.7845892868043%29%2A%28x-1.71541071319574%29
Again, the answer is: 10.7845892868043, 1.71541071319574. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-25%2Ax%2B37+%29

Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
Thomas is going to make an open-top box by cutting equal squares from the four
corners of an 11 inch by 14 inch sheet of cardboard and folding up the sides.
If the area of the base is to be 80 square inches, then what size square should
be cut from each corner?
:
Let x = side of the square to be cut off
:
Box dimension : (11-2x) by (14-2x) by x
:
Base as given = 80 sq/in, therefore
(11-2x) * (14-2x) = 80
FOIL
154 - 22x - 28x + 4x^2 = 80
:
154 - 50x + 4x^2 - 80 = 0
Arranged as a quadratic equation:
4x^2 - 50x + 74 = 0
Simplify divide by 2
2x^2 - 25x + 37 = 0
Solve this using the quadratic formula:
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
in this equation a=2, b=-25, c=37
x+=+%28-%28-25%29+%2B-+sqrt%28-25%5E2+-+4+%2A+2+%2A+37+%29%29%2F%282%2A2%29+
:
x+=+%2825+%2B-+sqrt%28625+-+296+%29%29%2F%284%29+
:
x+=+%2825+%2B-+sqrt%28329+%29%29%2F%284%29+
Two solutions
x+=+%2825+%2B+18.14%29%2F%284%29+
x+=+43.14%2F4
x = 10.785
and
x+=+%2825+-+18.14%29%2F%284%29+
x+=+6.86%2F4
x = 1.715 inches (the solution that makes sense)
;
:
Check solution
(11-2(1.715)) * (14-2(1.715)) =
(11-3.43) * (14-3.43) = 80.0