Which way can you answer the problem?

Question

An open box is to be made from a rectangular piece of material by cutting equal squares from each corner and turning up the sides. Find the dimensions of the box of maximum volume if the material has dimension of 2 feet by 3 feet.
answer: 5-sqrt(7)/6ft. by 1+sqrt(7)/3ft. by 4+sqrt(7)/3ft.
I have a answer, but I want to learn how to solve it. This is a calculus problem.

Answer 1

You should draw a picture. The 2x3 sheet is divided into 9 parts. If H is the height, L the longest base, and W, the other base, there will be 4 HxH pieces removed, and the box will have the dimensions L,W,H. The produce of these is the volume, which we want to optimize. So Volume= L*W*H, but we have to choose one variable to work with of the three. If you look at the sheet,
L+2H= 36, and W+2H=24. So we can express everything in terms of H:
V = H x 36-2H x 24-2H= H x [864-120H+4H^2]
Well, you will wind up with a cubic in H. When you take the derivitive dV/dH, you will wind up with a quadratic. Only one of the roots for H will be realistic. Then you can go back and determine the other sides. BTW: H IS [5-sqrt(7)]/6 ft,

Answer 2

The best way to solve this problem is to draw a picture of a rectangle with dimensions of 2 feet by 3 feet, then drawing a box shape in each corner to show how it is cut. Label the side of the box cutout with x. From there you should notice the 3 measurements you need to find volume of a rectangular prism:
Length = 3-2x
Width = 2-2x
Height = x
The volume of the prism is now:
V = (3-2x)(2-2x)x = 6x -6x^2 - 4x^2 + 4x^3
V = 4x^3 - 10x^2 + 6x
We now need to find out where the extreme values are so we can pick the extreme value when volume is the highest. Take the derivative to find this:
dV = 12x^2 - 20x + 6
Set this equal to 0 and solve to find the maximum
12x^2 - 20x + 6 = 0
After using the rule for solving a quadratic I get (20+sqrt112)/24 and (20-sqrt112)/24 but the latter makes more sense to use
So the dimensions are now:
Length = 3-2((20-sqrt112)/24)
Width = 2-2((20-sqrt112)/24)
Height = (20-sqrt112)/24
Those should simplify to your answers. I hope this helps.

Answer 3

Assume that the square to be cut are of size y by y (y x y)
Now if you make a box out of it:
Its height is y
Length is 3-y ft
Width is 2-y ft
Volume = y(2-y)(3-y)
=(2y- y^2)(3-y)
= y^3 - 5y^2 + 6y
To maximize the volume differentiate the equation and equate it to zero
3y^2-10y+6=0
Solve the equation and you will get the answer.
(Double check the calculations)