Question

You need to form a 10-inch by 15-inch piece of tin into a box (with no top) by cutting a square from each corner and folding up the sides. how much should you cut so the resulting box has the greatest volume?

Answer 1

This sort of question is answered easily by a graphing calculator. The square cut from each corner should be 1.962 inches on each side.

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After creating a fold-up flap of x inches in width, the base of the box will be
  (10 - 2x) by (15 - 2x)
and the depth of the box will be the width of the fold-up flap: x.

Then the volume of the box is
  v = x(10 -2x)(15 -2x) = 150x -50x^2 +4x^3
The derivative of the volume will be zero at the maximum volume.
  0 = dv/dx = 150 -100x +12x^2
This has roots at
  x = (100 ±√(100² - 4(12)(150)))/(2·12)
  x = (100 ± √2800)/24 = (25 ± 5√7)/6
Only the smaller of these solutions gives a maximum volume.

You should cut (5/6)(5-√7) ≈ 1.962 inches to obtain the greatest volume.