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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.
Volum V = L * W * H
square baset --> W = L =10
So V =10 *10 * L
V =100L
Its dimensions are20/3 and20/3 and5/3