page+328+problem+6

 very good! Ms. Sweeney

6) Find the dimensions of a rectangle with area 1000 whose perimeter is as small as possible.

first I started by finding equations I know: objective function --> what is trying to be max/min? perimeter. Perimeter = 2W + 2L. because this is our objective we know that it will be what we are taking the derivative of. Then I find my other equation, the one that will be minipulated to fit into our objective equation with given information. We are give area. A = LxW and in our case it is LxW = 1000.

Then we solve for 1 variable. L = 1000/W (wow I'm good). and put that guy back into our objective.

We now have, Perimeter = 2W + (2) (L) --> L can be substituted.... Perimeter = 2W + 2 (1000/W)

yay! we have an equation to derive! what are we looking for? oh, the minimum...relative minimum of this equation!

so we derive it and set it = 0

F(x) = 2W + 2000/W

F'(x) = 2 - (2000/w²) ........ (low d high...yadayada)

set it = 0

0 = 2 - 2000/w²

2 = 1000/w²

w = squrt (1000)

distance can't be negative so w = +31.6227766 (ish)

let's see, do we have a relative min?

---2-31.6227766130928473- (number line)

at 2, derivative is negative, at 130928473 derivative is positive. f(x) is decreasing then increasing because f'(x) is negative to possitive (yaddayadda) --> relative min. yay.

so, what are we asked for here? We want the dimensions.... so let's plug in.

1000 = LxW 1000 = L x 31.6227766 = 31.633

woah, it's a square, L = W who would've guessed?

dimensions are 31.6227766m, x 31.6227766m (ish)

But now lets make sure that this the smallest possible perimeter: There are no restrictions on the amounts of units available for the perimeter, therefore it is possible for either L or W to be very close to 0 or for either L or W to be very very long. We cannot have a negative distance, so they must be positive. Therefore domain for W is (0, infinity)

We now have to take the limit as W approaches each end point. first 0

lim 2W + 2 (1000/W) = infinity w-0

lim 2W + 2 (1000/W)= infinity w-infinity

both infinity and infinity are larger than the dimensions 31.62 by 31.62 therefore the smallest possible dimensions are 31.6227766m, x 31.6227766m (ish).