A SHEET OF CARDBOARD 180 INCHES SQUARE IS USED to make an open box by cutting squares of equal size from the corners and folding up the sides, what size squares should be cut to obtain a box with the largest possible volume?

Answer 1

If x represents the side of each of the 4 little squares cut from the corners, then the dimensions of the box would be (180-2x), (180-2x) and x.
Volume,
V(x)=x(180-2x)²
Find V'(x)=dV(x)/dx and equate to zero. Solve for x. Retain only positive root, x0.
To make sure the value of x is maximum, Find V"(x)=d²V(x)/dx² and evaluate V"(x0).
If V"(x0)<0, V(x0) is at its maximum.
Post your answer for a check if you wish.

Answer 2

180