What are the dimensions of the box with the largest volume?

Question

You are constructing a cardboard box with the dimensions 2 m by 4 m. You then cut equal-size squares from each corner so you may fold the edges.

please do it step by step because I don't understand this topic

Answer 1

Mustafa

You start with a piece of cardboard 2m x 4m

You cut a square dimensions l x l m from each corner and fold the resulting cardboard up to form an open box (with no top)

You need to find l, so that the volume is maximum

The box formed will have length (4-2l), width (2-2l) and height l, so

volume V = l (4-2l) (2-2l)

Volume will be maximum/minimum when dV/dl = 0

Maximum if d²V/ dl² < 0

So V = l (8 - 12l + 4l²) = 8l - 12l² + 4l³

dV/dl = 8 - 24l + 12l²

^{The second derivative is -24 + 24l}

dV/dl = 0 when 8 -24 l + 12l² = 0

You can solve this using the quadratic formula or plotting it on Desmos and finding the zeroes.

Giving x= 0.423 or x = 1.577 (x must be <=1 so this is an invalid solution).

When x = 0.423 the second derivative is negative, so that gives the maximum volume

So solution is : cut squares size 0.423 x 0.423 m from each corner

The resulting box has volume 1.54 cubic meters

(another way not involving differentiation is to just plot the V function on Desmos and see which x gives highest value of y. You only consider 0 < x < 1).