Question

A box without a top is to be made from a rectangular piece of cardboard, with dimensions 8 in. by 10 in., by cutting out square corners with side length x and folding up the sides.

Answer 1

...haha, "what is the height x, which produces the box with the maximum volume"

V=x(8-2x)(10-2x)  (note here that x<4 and x>0 for real solutions)

V=x(4x^2-36x+80)

V=4x^3-36x^2+80x

dV/dx=12x^2-72x+80

d2V/dx2=24x-72

The maximum volume will occur when the velocity dV/dx=0 and the acceleration d2V/dx2<0.

dV/dx=0 when:

12x^2-72x+80=0, using the quadratic formula for expediency:

x=(72±√1344)/24, since 0<x<4

x=(72-√1344)/24 in

x≈1.47 in (to nearest hundredth of an inch)

The maximum volume would be:

V((72-√1344)/24)=4x^3-36x^2+80x

V≈52.51 in^3  (to nearest hundredth of a cubic inch)