Hi can someone help me with this quest? It's a optimization question.

Question

anonymous · Answer

This is the question in my text book, and the answer is supposed to be $$10\sqrt{2}ft \times 40\sqrt{2ft}$$

anonymous · Answer

did u draw the figure?

anonymous · Answer

|dw:1417470548563:dw|

anonymous · Answer

The management of the UNICO department store has decided to enclose an 800ft^2 area outside the building. One side will be formed by the external wall of the store, two side will be constructed of pine boards, and the fourth side will be made of galvanized steel fencing. If the pine board fencing costs $6/running foot and the steel fencing costs $3/running foot, determine the dimensions of the enclosure that can be erected at minimum cost.

anonymous · Answer

If P=Pine and S=Steel

then 

$$P \times S = Area$$

and $$2p +s = Perimeter $$

anonymous · Answer

Im not sure how to differentiate this.

anonymous · Answer

Surely the cheapest option will be a 1x800ft area?
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The pine is the most expensive resource, so we want to use as little as possible.

anonymous · Answer

Not exactly. I posted the answer is $$10\sqrt{2ft} \times 40\sqrt{2ft}$$ in the textbook

anonymous · Answer

Oh crap, I've really messed this one up. Okay, I'll get differentiating haha, back soon!

anonymous · Answer

Haha no worries, than you. I'm really stuck because I don't know what to do the with units $$ft^2$$ when I differentiate.

I would like to know for future problems :)

anonymous · Answer

$$P \times S = 800ft^2$$

$$S = \frac{ 800ft^2 }{ P }$$

Thus $$2P +\frac{ 800ft^2 }{ P } = Perimeter$$

anonymous · Answer

$$f(p) = 2p + 800ft^2 \times P ^{-1}$$

Is this correct?

anonymous · Answer

Ditch the units, you don't need to consider them :) I have it written slightly different, but it's the same: $$ f(p)=\frac{2p^2+800}{p}$$

anonymous · Answer

When I differentiate that and equate it to 0, I get p=20 and I keep getting this answer no matter what I try. Hmm

anonymous · Answer

Is is exactly what I keep getting and I wish someone could explain how to get $$10\sqrt{2ft} \times 40\sqrt{2ft}$$

anonymous · Answer

I've done it!

anonymous · Answer

We've been minimising the wrong thing completely, our condition is PS=800 and we were previously using A=S+2P and minimising A, but we should be putting the prices of each into this and instead minimising A=3S+12P. With the same approach as before, you get the correct answers :)

anonymous · Answer

Yes! Finally I get it. Thank you!

anonymous · Answer

No problem! I can't believe we were leaving out the most important piece of information given to us haha