Max/Min problem I'm fundamentally flawed in my understanding I think

[monet A]

I am entirely frustrated with maximum and minimum problems. I have had this issue with them since I first was introduced to them and I thought I had resolved it but it keeps getting up and biting me again.

The problem is I am not even sure what I am in the habit of doing wrong, so I can't fix it. Could somebody point out what I have done ar*eways here for me so I can get over this once and for all..?

A closed box is to be made with length equal to 3 times its width. The total surface area of the box will be 30m^2. Find the dimensions that give maximum volume in the box.

$$3000cm^2 = 2lw + 2hw + 2lh$$

$$\frac {3000 - 6w^2} {8w}} = h$$

$$V = l *w * h$$

$$V(w) = 3w * w * \frac {3000 - 6w^2} {8w}} = \frac {9000w - 18w^3} {8}}$$

$$V'(w) = \frac {9000 - 54w^2} {8}}$$

$$V''(w) = \frac {- 108w} {8}}$$

Now when I find the 0 value of dv/dw it is one value that can be positive or negative, naturally I disregard the negative value, but then plugging anything positive into the second derivative I will always get a negative slope, so all values >0 would return as a maximum. I can finish off the problem from there and get values.

However...

My problem is that if I work it out in units of metres by allowing 30m^2 to equal the surface area I get a value of ca 1.29 metres, and if I use cm^2 (above) as the surface area unit I get ca 12.9cm. Both work to give 30m^2 as the surface area but they are clearly not the same values yet they are working perfectly in the same equation?

Should I have found both of these as critical points when looking for 0 in my 1st derivative? If so, how, what did I miss...

please someone help me iron out this crinkle in my understanding because as you can see I am too confused to do it myself.

 

[Galileo]

30 square meters is not equal to 3.000 square centimeters but 300.000 square centimeters.

$$1 \mbox{m}^2 = 10.000 \mbox{cm}^2$$

you have to reduce both dimensions by a factor of 100.

 

[monet A]

30 square meters is not equal to 3.000 square centimeters but 300.000 square centimeters.

$$1 \mbox{m}^2 = 10.000 \mbox{cm}^2$$

you have to reduce both dimensions by a factor of 100.

Oh as in 100 cm * 100 cm = 10000 cm^2 I thought it was a fundamental flaw in my understanding but *that* fundamental is just embarrassing, thanks for pointing it out to me I think
I am sure I won't let this happen again, I might get paranoid enough to question all my chemistry calculations though... what have I done?

 

[trancefishy]

don't worry, if you pay attention, you will see that people make this mistake all the time. pretty much any news article you read that converts units^2 does it incorrectly.

to make things worse even, in my 300 level geography class a few weeks ago, the instructor was saying something about how many square kilometers of rainforest were destroyed each year. I think it was like, 25,000km^2 were destroyed. he's like "anyone got a calculator, so we can put that in miles?". and this nerdly kid calculates it, and comes up with 14,000mi^2 (which is wrong, and dumb to boot, since he had a ti-8x with the conversion built in..). to make things worse, the instructor says "so, that's about 7,000miles by 7,000 miles."

yup, good thing I'm at a high quality university.

 

[quasar987]

lol fishhy

monet: A good trick not to get confused about unit conversion is to do this: 1m = 100cm, so

$$30m^2 = 30 (100 cm)^2 = 30\cdot 100^2 \cdot cm^2 = 300000cm^2$$

cool huh?

So as soon as you know the "linear conversion", you can retrieve any other "higher degree" conversion. Another example, taken from physics: 1eV = 1.6*10^-19 J, so

$$5\cdot 10^9eV^3 = 5\cdot 10^9(1.6\cdot 10^{-19} J)^3 = 5\cdot 10^9\cdot (1.6\cdot 10^{-19})^3 J^3 = 8\cdot 10^{-10} J^3$$