SOLUTION: A rectangular piece of cardboard 11 in by 14 in is made into a box by cutting identical squares from each corner and folding up the sides. If the bottom of the box turns out to hav

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Question 242992: A rectangular piece of cardboard 11 in by 14 in is made into a box by cutting identical squares from each corner and folding up the sides. If the bottom of the box turns out to have an area of 80 sq. in. what size squares were cut from the corners?
Answer by JimboP1977(311) About Me  (Show Source):
You can put this solution on YOUR website!
The best way to do this is to draw a diagram and mark in dashed lines the corner sections and centre section.
The whole area of the cardboard is equal to 154 inches^2. The corner sections are equal to 4x^2. The mid sections are equal to 2x(14-2x) and 2x(11-2x).
So we know that the total area of the cardboard minus the 4 corners minus the 4 mid sections must equal the centre bottom area of 80 inches^2.
154-4x%5E2-2x%2814-2x%29-2x%2811-2x%29+=+80
154-4x%5E2-%2828x-4x%5E2%29-%2822x-4x%5E2%29+=+80
4x%5E2-50x%2B74+=+0 Rearrange to equal zero.
Use the quadratic formula to find x.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B-50x%2B74+=+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=%28-50%29%5E2-4%2A4%2A74=1316.

Discriminant d=1316 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--50%2B-sqrt%28+1316+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-50%29%2Bsqrt%28+1316+%29%29%2F2%5C4+=+10.7845892868043
x%5B2%5D+=+%28-%28-50%29-sqrt%28+1316+%29%29%2F2%5C4+=+1.71541071319574

Quadratic expression 4x%5E2%2B-50x%2B74 can be factored:
4x%5E2%2B-50x%2B74+=+4%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+4%2Ax%5E2%2B-50%2Ax%2B74+%29


x cannot be 10.784 as this would mean it was longer than the sides which of course does not make sense! So it must be 1.7154 inches. So the size of the squares cut was 1.7154^2 = 2.942 inches^2.