Task 1: What are the dimensions of the largest volume of the box, with a square base, open at the top, formed of a square of a = 10 cm?

When x is the length of the sides of the squares removed, it is the height of the box, and its base has a field (10 - 2x) 2 Thus, it suffices to find a function f (x) = (10 - 2x)2. x. Maybe instead of exploring the function, just to see her chart? (Then it will be available not only for knowing derivatives.) This approach has drawbacks:     - Graph of f is very 'tight' (it may be better to use an auxiliary function g?)     - A computer (not a student) put a dot, found extreme features,     - It is difficult to design routing functions, which inscripturated ignored here,     - And even the little things: why such signs, why f rather than V, why x, and not h?

Thank you Tomasz Modrzejewski for inviting me to answer this question but I see that I'm late and the question is already answered. the dimensions are20/3 and20/3 and5/3 as one of the people have answered and the method they used is correct.

The question it seems is not quite clear. Let me clarify.  A box without top is to be formed of a cardboard piece of10cm by10 cm. What will be box's dimensions if it has largest volume of all such boxes.

Now I shall answer it.

Fold it x cms from all sides, and eureka, the box is formed.

Now the volume of this box is (10-2x)*(10-2x)*x=100x-40x**2+x**3

For this to be maximum or minimum, its derivative has to be equal to zero.

so100-80x+12x**2=0

or25-20x+3x**2=0

or (5-x)(5-3x)=0

or x=5, x=5/3

at x=5, the volume is zero i.e minimum.

 hence at x=5/3, the volume is  maximum.

So dimensions of box at maximum volume are20/3cm,20/3cm,5/3cm.

strictly the box is open, so there is no Volume enclosed

Let x=be the height(H) of the cube

So with x as height, we can get10-2x as the length(L) and width(W)

Therefore, V(Volume) = L W H

V = (10-2x)(10-2x)(x)

V = (100-40x+4x2)(x)

V =100x-40x2+4x3

get V prime,

V'=100-80x+12x2

V'=(20-4x)(5-3x)

from this we can get values of x,

x =5 and x=5/3, correct value for x is5/3 since if x=5, the value of length and width will be zero.

So, height which is x=5/3, we can compute the value of length and width:

length or width =10-2x =10-2(5/3)

                        =20/3

Therefore, dimension of the cube with a largest Volume is20/3 x20/3 x5/3.

length=10cm.

width=10cm.

but height we can't say,because not mention the box height.

even though box base is square,the box heigth can be vary.

30 cm

100 CM

Volum V = L * W * H

square baset --> W = L =10

So V =10 *10 *  L

V =100L

ANS IS20/320/35/3

Its dimensions are20/3 and20/3 and5/3