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I am suppose to find the volume if 1200 cm^2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

I think what I need to do is set up the formulas to be

$4(lw) + w^2 = 1200$ for area

$lwh = v$ for volume

I know that if the base is a square than the rectangle will have the same dimensions and the only different variable would be the height so I can solve for length like so

$l=\frac{1200-w^2}{4w}$

Now that I have that I can put it in my formula

$lwh = v$ for volume

which I can rewrite as

$l^2 * h = v$

I then take the derivative of this and I get some ridiculous answer that is wrong.

$\frac{1200w^2 - w^4}{4w}$

the derivative

$300 \frac{-3w^2}{4}$

Which gives me $+-20$ which is an incorrect answer.

2 Answers 2

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The formula $\rm V=lwh$ means "volume = length times width times height." The variable $\rm l$ is length, the variable $\rm w$ is width, and the variable $\rm h$ is height. Using these, the total area is actually

$$\rm 2(l\times h)+2(w\times h)+w^2=1200.$$

We know that $\rm l=w$ (because the base of the box is square), so this is $\rm 4wh+w^2=1200$. This allows us to solve for the height $\rm h$ in terms of width $\rm w$ as $\rm h=(1200-w^2)/(4w)$.

We have the formula for $\rm h$ in terms of $\rm w$, and know $\rm l=w$, so we have the volume function

$$\rm V=l\,wh=w^2\frac{1200-w^2}{4w}=300w-\frac{1}{4}w^3.$$

Now can you take the derivative of this, equate it with zero and solve for $\rm w$?

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    The zero is 20 but I am not sure what to do with it or what it means anymore.I think I got it now, I just substitute 20 into the other formula and then I find that the height is 10 and put all that into the volume formula which gives me 4,000.2012-04-02
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    @Jordan: As per the [derivative test](http://en.wikipedia.org/wiki/First_derivative_test) (and a couple other checks), this means the volume is optimized when the width is twenty. Plug $\rm w=20$ into the formula for $V$ and you will have the optimized volume.2012-04-02
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When you write down a formula, you must write down what the letters stand for.

When you write down $4lw+w^2=1200$, you must add, "where $l$ is the length of the base of the box, and $w$ is the width of the base of the box".

If you do that, you might see right away where you've messed up. You've written down a formula for the area which doesn't include the height of the box---that can't possibly be right, can it?

In fact, the formula you have written down only makes sense if $l$ is the height of the box, right?

So go back and identify the variables explicitly and then write down formulas that make sense.

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    I don't really know, I have trouble keeping track of all these formulas I need to memorize and all these rules to doing these types of problems. There are just so many parts I need to keep track of that I always mess up a few of them.2012-04-02
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    For some reason I keep getting a derivative of $\frac{48002^2 - 12w^4}{4w^2}$ which is ofcourse wrong.2012-04-02
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    To add to Gerry's great suggestions, I would recommend drawing a good figure of your box and label it with all of your variables. This is another way to catch your error.2012-04-02
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    I did that, I have been working on this problem for over an hour now but I can't get the proper answer there are just too many parts for these problems where I will always mess up at least one thing and I never know which one I mess up so I redo the whole problem every time and I always get different answers.2012-04-02
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    What variables are you using for the sides of your base square, and what variable are you using for the height of the box?2012-04-02
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    Gerry, I don't believe the labeling error you identify is causing the real problems Jordan has.2012-04-02
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    The height of the box is l and the sides of the base is w.2012-04-02
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    If the height of the box is $l$, Jordan, then what is the $h$ that you have written down?2012-04-02
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    @robjohn, I've interacted with Jordan before, and I know something about the real problems. I despair of doing much about them, but maybe if I can convince Jordan not to write down formulas without also writing down what the letters mean I can be of some help.2012-04-02
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    I am also familiar with Jordan, I just wanted to note there was an issue besides the incorrect labels. (Also, for future note: I am actually *anon*, but through an ill-planned April Fools joke I am carrying robjohn's name for the moment.)2012-04-02
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    @robjohn (or anon, or whatever), I take your point. I propose to deal with one issue at a time.2012-04-02