Optimalization problems

[lamaclass]

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?

 

[BigGlenntheHeavy]

$\displaystyle lamaclass, \ did \ you \ review \ you \ previous \ thread \ below \ "Finding \ Relative \ Extrema"?$

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

$\displaystyle \ was \ wrong, \ as \ it \ isn't.$

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

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

$\displaystyle admitting \ his \ mistake.$

 

[lamaclass]

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!

 

[lamaclass]

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!

 

[BigGlenntheHeavy]

$\displaystyle 2) \ V_{cyl} \ = \ \pi r^{2}h, \ A_{cyl} \ = \ 2\pi rh+\pi r^{2}$

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

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

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

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

 

[lamaclass]

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?

 

[lamaclass]

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. :?