calculus, area, dimensions, height of box

Question

An open box is to be made out of a 8-inch by 16-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

Dimensions of the bottom of the box:  

Height of the box: 

Answer 1

By V (x) = 128x - 48x^2 + 4x^3, the derivative V ' (x) = 4 * ( 3x^2 - 24x + 32).

Setting V ' (x) = 0 (no need to worry about DNE senarioes because this polynomial does not have such situation) and solving for x = 4 ± (4/3) sqrt(3). Checking the derivative signs before and after x ≈ 1.69 tells that this x is at a relative maximum, and checking the derivative signs before and after x ≈ 6.3 tells that this x is at a relative minimum (discard this now). Therefore, when x = 4 - (4/3)sqrt(3) yields the volume at its maximum.

So, Length is about 12.6, and the Width is about 4.6. The height is just x itself! Hope this helps!