Optimalization problems

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lamaclass

Junior Member
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Oct 18, 2009
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Hi, I was wondering if someone could show me how you would set these two problems up? Thanks! :)

1) A landscape architect wishes to enclose a rectangular garden on one side by a brick wall costing $30/ft. and on the other three sides by a metal fence costing $10/ft. If the area of the garden is 1,000 square ft, find the dimensions of the garden that minimizes the cost.'

2) What is the maximum possible volume of a cylindrical can with no top that can be made from 27 pi square inches of metal?
 
\(\displaystyle lamaclass, \ did \ you \ review \ you \ previous \ thread \ below \ "Finding \ Relative \ Extrema"?\)

\(\displaystyle You \ wanted \ to \ know \ why \ 0 \ wasn't \ a \ critical \ point \ and \ Subhotosh \ Khan \ said \ it \ was, \ but \ he\)

\(\displaystyle \ was \ wrong, \ as \ it \ isn't.\)

\(\displaystyle This \ is \ one \ reason \ people \ get \ disgusted \ with \ math, \ as \ they \ are \ fed \ the \ wrong \ information \ by\)

\(\displaystyle \ people \ who \ should \ know \ better \ and \ when \ the \ culprit \ is \ exposed, \ he \ remains \ silent, \ instead \ of\)

\(\displaystyle admitting \ his \ mistake.\)
 
I did look at it once before previously and just saw now that more information was added to it that explains furthur. I will read through it to better understand it, thank you!
 
I'm still uncertain on how to first approach these problems though. If anyone could show me how to set them up, it'd be greatly appreciated! :D
 
\(\displaystyle 2) \ V_{cyl} \ = \ \pi r^{2}h, \ A_{cyl} \ = \ 2\pi rh+\pi r^{2}\)

\(\displaystyle Ergo, \ 27\pi \ = \ 2\pi rh+\pi r^{2}, \ 2\pi rh \ = \ 27\pi-\pi r^{2}, \ h \ = \ \frac{27-r^{2}}{2r}\)

\(\displaystyle Then, \ V_{cyl} \ = \ \pi r^{2}\bigg(\frac{27-r^{2}}{2r}\bigg) \ = \ \frac{27\pi r-\pi r^{3}}{2}\)

\(\displaystyle Hence, \ \frac{dV}{dr} \ = \ \frac{27\pi-3\pi r^{2}}{2} \ = \ 0 \ \implies \ 3\pi r^{2} \ = \ 27\pi, \ \implies \ r \ =3, \ ergo, \ h \ =3\)

\(\displaystyle Therefore, \ V_{cyl} \ = \ \pi(3)(9) \ = \ 27\pi \ in^{3} \ = \ Max.\)
 
Thanks Glenn!

So for the first one now, to start off with, would my equation for it be A=10x[sup:aoum3v41]3[/sup:aoum3v41]*30y with 1000 as the constraint so A'=30x[sup:aoum3v41]2[/sup:aoum3v41]*30y dy/dx?
 
Am I on the right track with this problem? I see that the constraint is 1000 but the 3 x along with the $10 I think is what's goofing me up. :?
 
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