The box problem

[Anguirel]

I need to do a PA for maths and I'm a bit stuck.
The PA is about folding a box with a volume that is as big as possible. The first few questions where really easy but then this one came up.

These are the questions that I have to answer:
1. Calculate the exact measurements in mm of A4 paper.
2. Same size squares are cut from each corner of an A4 paper. After that a basket is folded. Find out what the size of the squares must be for the volume to be maximum.
3. Find out the maximum using a graph.
4. Find out the maximum by differentiating.
5. Same task as at 2. Start with a square piece of paper of 15 by 15 cm.
6. Choose at least another two square sizes of paper and determine the maximum volume of those sizes.
7. Start with a square paper with sides of z cm by z cm. Make another basket with a maximum volume. Find a formula to express the maximum content in z.
8. Prove by differentiating that the formula at 7 does indeed give you the maximum volume for each value of z.

So I already have the answers for questions 1 to 7 but I thought that it would maybe be useful to have an overview of the PA. My question is about number 8. For that you need question 7 so here are my calculations for 7:

V = l x h x w
l = w = z - 2h

V = h(z - 2h)^2
V(h) = h(z^2 - 4hz + 4h^2)
V(h) = 4h^3 - 4zh^2 + (z^2)h

So with the last formula you can calculate the volume and by using a graph you can read off the maximum volume. Or you can use the derived function: V'(h) = 12h^2 - 8zh + z^2 = 0


That is my answer to question 7. But how do you prove (by differentiating) that it's true for all values of z?
I tried to do something with the fact that z/h = 6 (I can send the calculations if anyone wants to). But I don't really see what use it has.

Any help is very welcome! Don't feel any pressure please but I kinda need to have it before coming wednesday...

 

[wjm11]

I need to do a PA for maths and I'm a bit stuck.
The PA is about folding a box with a volume that is as big as possible. The first few questions where really easy but then this one came up.

These are the questions that I have to answer:
1. Calculate the exact measurements in mm of A4 paper.
2. Same size squares are cut from each corner of an A4 paper. After that a basket is folded. Find out what the size of the squares must be for the volume to be maximum.
3. Find out the maximum using a graph.
4. Find out the maximum by differentiating.
5. Same task as at 2. Start with a square piece of paper of 15 by 15 cm.
6. Choose at least another two square sizes of paper and determine the maximum volume of those sizes.
7. Start with a square paper with sides of z cm by z cm. Make another basket with a maximum volume. Find a formula to express the maximum content in z.
8. Prove by differentiating that the formula at 7 does indeed give you the maximum volume for each value of z.

So I already have the answers for questions 1 to 7 but I thought that it would maybe be useful to have an overview of the PA. My question is about number 8. For that you need question 7 so here are my calculations for 7:

V = l x h x w
l = w = z - 2h

V = h(z - 2h)^2
V(h) = h(z^2 - 4hz + 4h^2)
V(h) = 4h^3 - 4zh^2 + (z^2)h

So with the last formula you can calculate the volume and by using a graph you can read off the maximum volume. Or you can use the derived function: V'(h) = 12h^2 - 8zh + z^2 = 0


That is my answer to question 7. But how do you prove (by differentiating) that it's true for all values of z?
I tried to do something with the fact that z/h = 6 (I can send the calculations if anyone wants to). But I don't really see what use it has.

Any help is very welcome! Don't feel any pressure please but I kinda need to have it before coming wednesday...

Your first equation, V(h) = 4h^3 - 4zh^2 + (z^2)h , is a general volume formula but does not give a maximum or minimum.

Your derivative equation, V'(h) = 12h^2 - 8zh + z^2 = 0 , when set to zero, allows you to find locations of local maxima or minima. You need to solve this for h using the quadratic formula. Plug the result back into your first equation to obtain a formula that gives either a maximum or a minimum. (In general, you must also check end points/boundary conditions/problem parameters for extrema. For example, in this problem, both h and z must be greater than zero, and h must be less than z. So if we plug in zero for h, our volume is zero, a minimum.)

Hope that helps.

 

[Anguirel]

Your first equation, V(h) = 4h^3 - 4zh^2 + (z^2)h , is a general volume formula but does not give a maximum or minimum.

Your derivative equation, V'(h) = 12h^2 - 8zh + z^2 = 0 , when set to zero, allows you to find locations of local maxima or minima. You need to solve this for h using the quadratic formula. Plug the result back into your first equation to obtain a formula that gives either a maximum or a minimum. (In general, you must also check end points/boundary conditions/problem parameters for extrema. For example, in this problem, both h and z must be greater than zero, and h must be less than z. So if we plug in zero for h, our volume is zero, a minimum.)

Hope that helps.

First of all: thanks! But I still don't get it

If I use the quadratic formula I get h=(1/6)z but I don't see the connection between that and the question. I mean, how does it prove that my formula gives the maximum volume for every value of z?
And even it can be proved in this way, it's a bit odd. For I use the formula which I have to prove for my proof (if you understand what I mean)...

Any ideas?

 

[wjm11]

First of all: thanks! But I still don't get it

If I use the quadratic formula I get h=(1/6)z but I don't see the connection between that and the question. I mean, how does it prove that my formula gives the maximum volume for every value of z?
And even it can be proved in this way, it's a bit odd. For I use the formula which I have to prove for my proof (if you understand what I mean)...

Any ideas?

You still need to plug h = z/6 into your volume equation to get the maximum volume.

THEN, you still need to show that it is indeed a maximum, and not a minimum or point of inflection. What does the second derivative tell us? What does it mean when the second derivative is positive, negative, or zero at a given value of h???