SOLUTION: By cutting squares out of the corners of an 8 1/2 in x 11in sheet of paper, and folding up the edges, you can form a box. Whats the largest possible volume of the box if the length

Algebra ->  Volume -> SOLUTION: By cutting squares out of the corners of an 8 1/2 in x 11in sheet of paper, and folding up the edges, you can form a box. Whats the largest possible volume of the box if the length      Log On


   

Question 1011095: By cutting squares out of the corners of an 8 1/2 in x 11in sheet of paper, and folding up the edges, you can form a box. Whats the largest possible volume of the box if the length cut is no less than 1in and no more than 4 in?
Found 2 solutions by stanbon, josgarithmetic:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
By cutting squares out of the corners of an 8 1/2 in x 11in sheet of paper, and folding up the edges, you can form a box.
Whats the largest possible volume of the box if the length cut is no less than 1in and no more than 4 in?
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Volume = x(8 1/2 - 2x)(11-2x)
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V = x[8.5-2x][11-2x]
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V = x[93.5 -17x -22x + 4x^2]
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V = 93.5x - 39x^2 + 4x^3
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V' = 93.5 - 78x + 12x^2
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Solve:: 12x^2 - 78x + 93.5 = 0
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x = 4.9146
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Cheers,
Stan H.
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Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
Sidelength of the squares cut, x.
v for volume, v=x%288.5-2x%29%2811-2x%29;
how? Height x multiplied by the area of base formed (8.5-2x)(11-2x).

v=x%288.5%2A11-22x-19x%2B4x%5E2%29
v=4x%5E3-%2822%2B19%29x%5E2%2B8.5%2A11x
highlight%28v=4x%5E3-41x%5E2%2B93.5x%29------general volume formula

FIND MAXIMUM
dv%2Fdx=12x%5E2-82x%2B93.5------derivative---BUT NOT NECESSARY....(?)

Extremes in volume would occur at x=%2882%2B-+sqrt%2882%5E2-4%2A12%2A93.5%29%29%2F%282%2A12%29
x=%2882%2B-+sqrt%282236%29%29%2F%282%2A12%29
x=%2882%2B-+sqrt%282%2A2%2A13%2A43%29%29%2F2%2A12%29
x=%2882%2B-+2sqrt%2813%2A43%29%29%2F2%2A12%29
x=%2841%2B-+sqrt%28559%29%29%2F12
These are 1.446 and 5.387. The larger of these makes no sense for the dimension of the paper. The 1.446 has some meaning. You could try second derivative to help check if this is a max or a min.

If you are allowed a graph tool,
graph%28400%2C400%2C-2%2C10%2C-2%2C40%2C4x%5E3-41x%5E2%2B93.5x%29

This looks like cut 1.446 inches at the corners will give the MAXIMUM volume.