A sheet of cardboard 25 cm by 40 cm will be made into an open-topped box by cutting equal-sized squares from each corner and folding up the four edges. what will be the dimensions of the box with the largest volume?

Answer 1

The classic question that just about every Calculus text uses as an example of
problems dealing with optimization.

Let the side of the cut-out-square be x cm
So dimensions of the finished box are 25-2x by 40-2x for the base and x for the height
V = x(25-2x)(40-2x)
= 4x^3 - 130x^2 + 100x
dV/dx = 12x^2 - 260x + 1000 = 0 for a max of V
3x^2 - 65x + 250 = 0
x = ...... use whatever method is your favourite to solve quadratics
remember 0 < x < 12.5 for the dimensions to make sense