Optimization

[mattflint50]

I have two problems that I couldnt figure out.
1.)A contractor is to fence off a rectangular field along a straight river, the side along the river requires no fence. What is the least amount of fencing needed to fence off 30,000 m^2 ?
2.)You have been asked to design an oil can shaped like a right cylinder to hold 2000 cm^2 of oil. What dimensions will use the least material?

Can someone please help.

 

[jacket81]

problem

Okay , the equation for lenght of fencing is 2w+L, and formula for area of fencing is wL = 30000.
So, L = 30000/w.

So, substitute this in first formula, 2w+(30000/w).
Differeniate it and set to zero, 2-(30000 w^(-2)) = 0.
Or, 2 = 30000w^(-2).
So, 2w^2 = 30000, or, w^2 = 15000.

Find squareroot of both sides, gives you w= 122.474 (approximately).
And L = 30000/122.474 = 244.95 (approximately).

Now for 2nd problem, first, I think that should be cm^3, instead of cm^2.
So, the formula for material is (2pi*r^2)+(2pi*r*h).
And formula for volume is pi*r^2*h=2000.
SO, h=2000/(pi*r^2).
Plug this in 1st formula, giving you (2pi*r^2)+(2pi*r*(2000/(pi*r^2)).
Differeniate and set to zero:
(2*2pi*r)-(2*2000*r^(-2)) = 0.
So, 4pi*r = 8000/r^2, or, r^3 = 8000/(4pi).
So, r = 8.6025 (approximately).
(If i didnt make any mistakes)