I am confused over my maths coursework, I am doing an investigation on the Open Box Problem. Where an open box is to be made from a sheet of card. Identical squares are cut off the four corners of the card. The card is then folded along the dotted lines to make the box. I understand this and i have looked at several kinds of squares with different lengths. I have also investigated rectangles, with different ratios. I have found a pattern, however what i need to find out is how differentiation is involved and a quadratic formula. I would be very thankful for you help.

Do you want to find the largest possible area enclosed by the box?
If so, the volume of the box is given by the base x height (obviously). The base should be square (as this gives the greatest area... you should be able to work out why). However, the issue here is to establish how big this square should be. If you are given a specified area of card, then:

Now, the height of the box is $X$, and so the volume is therefore:

$4X^3-(4\times \sqrt{A}\times X^2)+\mathrm{AX}$, which can then be differentiated. To find when this value is at its greatest, work out values of $X$ when $f(x)=0$, and then take the second derivative of the equation to check whether the values are minimums or maximums (as you should have been taught how to do). I hope this is a)helpful and b) (more importantly) correct!

You should be expected to know and be able to use differentiation, as Philip said, if you are studying maths for A levels or further.

However if you are talking about GCSE's ( which I think you might be ) then calculus will be unfamiliar. To learn about this I recommend you ask your teachers for a A level standard text book containing an introduction.

Thanks for you help, Philip and James. I am studying GCSE Higher level maths. I have already asked my teacher for an A level standard text book, however i am very confused of how to connect this to my problem. I need to determine the size of the square cut that makes the box as large as possible for any given rectangular sheet of card. I know that this involves differentiation, but i need to use a simple form of differentiation and connect this to my equation. I know that it will involve a quadratic equation. I need an ultimate formula that works for any square or rectangle.If any one knows how to work this out, in a way not too complicated though verging into simple A level work i would be very thankful. PS:I know that the area should be worked out by the Volume = (Length -2 x cut out) x (Width -2 x cut out) x (Height).

Ok, lets if I get it.
I called your length: a
width:b
and cut-out:r
notice that the height is also r
Look at the following (sorry excuse for a) drawing: (It is just for illustration)


So out volume is:
(height) x (length-2 x cut_out) x (width-2 x cut_out)
or:
V=r x (a-2r)(b-2r)
open it up:
V=r(ab-2rb-2ra+4r² )=rab-2br² -2ar² +4r³
=4r³ -(2a+2b)r² +rab
If you know how to draw this kind of function, then go ahead and find when it gets the heights value (biggest volume).
The easiest way, is by doing differentiation.
When the function of the differentiated original function has values of zero (the roots), then at this point the original function has a minimum or a maximum value.
The easiest form of differentiation (What you need to know for this problem) is that:
The derivative of axⁿ is nax^n-1 .
And another thing you need to know that if you have a function composed of some parts (each part is set apart from a different part by either a plust sign or a minus sign), you can differentiate each part by itself and add the differentiations.
So our function is (V is depended on r, so we say that we differentiate V by r)
V(r)=4r³ -(2a+2b)r² +rab
the derivatives are:
4r³ => 4 x 3 x r² =12r²
-(2a+2b)r² => -2(2a+2b)r=-4(a+b)r
rab => 1 x ab x r⁰ =ab
We add the parts and (V'(r) is the derivative of V(r) )
V'(r)=12r² -4(a+b)r+ab

Now, we want to know when this function, that we got, has roots (gets a value of zero), we do that using the quadratic formula.
Now you will get two solutions, (If one of them is negative, wipe it out, because r can't be negative), to find out which one is for the maximum volume and which is for the minimum, take the rs (plural) that you got and stick it in the original function ( V(r) ), and the bigger value is the Maximum volume of the box.

Hope it helped,

Yatir

Thank you so much Yatir. It really helped how you explained it and i now understand how to differentiate my problem

Thank you,

Melanie

Any Time!

Yatir

Don't forget that you need to declare any help you have had. This shouldn't affect your mark, as the work being explained is beyond GCSE level, provided you show in your write-up that you understand it. It's probably best to print out this discussion to attach to your coursework or show to your teacher.