An open box is made from a rectangular piece of cardboard of dimensions 16 x 10

Question

An open box is made from a rectangular piece of cardboard of dimensions 16 x 10 in. by cutting out identical squares from each corner and bending up the resulting flaps. Find the dimensions of the box with the largest volume that can be made. length in width in height in An open box constructed from a tin sheet has a square base and a volume of 234 in.^3. Find the dimensions of the box, assuming that the minimum amount of material was used in its construction. (Round your answers to one decimal place.) length in width in height in Find the dimensions of the shaded region so that its area is maximized. (Enter your answers as a comma-separated list.) sides Height

Explanation / Answer

1. If we cut a square out with a side length of x units from each of the four corners, then the result will be a box with a length of 16 - 2x units, a width of 10 - 2x units, and a height of x units.

Then, the volume of the box is:

V(x) = lwh = x(16 - 2x)(10 - 2x) = 4x^3 - 52x^2 + 160x.

By taking derivatives:

V'(x) = 12x^2 - 104x + 160 = 4(3x^2 - 26x + 40).

Setting V'(x) = 0 gives x = (26 ± 14)/6. x=20/3, 2

Then, with V''(x) = 24x - 104, we see that V''(x) < 0 when x =2and V''(x) > 0 when x = 20/3. Thus, x =2. Then the resulting box has demensions 12 inches by 6 inches by 2 inches.

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