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A box is made from a rectangular piece of metal whit length 20 inches and width 15 inches. The box has no top. What are the limitations on the size of x?
x<7.5
To find the maximum volume of the box:
A square of side x inches is cut from each corner of the sheet and the remaining flaps folded up to make the topless box.
The box therefore has a height of x", length of 20 - 2x and width of 15 - 2x.
Volume is thus x * (20 - 2x) * (15 - 2x) ie 300x - 70x2 + 4x3
The volume is greatest when this has a maximum value:
f(x) = 300 - 140x + 12x2 (or 12x2 - 140x + 300)
Quadratic formula gives a value within the limit of x = 2.83 representing a box 14.34 * 9.34 * 2.83 ie 379.04 cu in.
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