find the demensions of the largest open box?

Question

that can be made from teh a sheet of tin that is 24in by 24in by cutting equal squares from the corners and turning up the sides

If you guys could give me some insight into working out it would be great, i've tried to do it but i seem at a loss....

Answer 1

Call the side of the square you cut out x, then the box you make will have a square base of side (24 - 2x) and the height will be x.

The area of this box is then (24 - 2x)^2*x = 2x*(12 - x)^2. Call this a function of x, and we need to find the maximum.

2x^3 -48x^2 + 288x = f(x)

6x^2 - 96x + 288 = f'(x) = 0

x^2 - 16x + 48 = 0

(x - 12)(x - 4) = 0

So this has the 2 answers x = 12 and x = 4.

x = 12 gives a minimum (volume = 0), so x = 4 is a maximum, 1024
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I just looked at the above answers. JimBurnel made a careless error, the volume should be x(24 - 2x)^2, not x(24 - 2x). The rest of his logic is right, but the equation is wrong.

And Paritosh Vasava assumes by "largest" you want surface area, I assume you mean volume. Of course the largest surface area comes when the squares you cut out are of size 0, your box is then 24*24*0, area = 576, volume = 0.

Answer 2

Let x be the length of one side of the squares that you cut in each corner.

Then the height of the box when you fold up the edges will be x. The width of the box will be 24 - 2x (that is, the original 24, less the 2 squares that you cut out).

The volume of this box would be V(x) = x(24 - 2x) = 24x - 2x²

So then the goal would be to find the value of x that maximizes this function.

If you haven't had calculus, you can only do trial and error.

If you have had calculus, you take the derivative, set it equal to zero, and solve for x:

V'(x) = 24 - 4x = 0 ---> x = 6

So the dimensions would be 6in tall and 24-2(6) = 12in wide.

Answer 3

The area of sheet is 24*24=576
The area you cut when you cut corners is 4*8*8 = 256

So you remain with 576-256 = 320

And the surface area of the box is
2(height*width)+2(length*width)+2(length*height)

Being a cube the height, length, breadth are same(lets say its 'a' and here a is 8). Also, the area of each side is also the same, so we can reduce the formula of surface area to
6(a*a) = 6(8*8) = 384
But, we have to deduct area of one of the faces to make it open.
So, its would be 5(a*a)=5(8*8)=320

I dont know if i have made any fumbles, but they are worth if you find a right track for your answer.

All the best