Question

Find the length and width of a rectangle that has an area of 512 square feet and whose perimeter is a minimum.

Answer 1

so far i have xy=512 and 2x+2y=P, so 2(512/y)+2y=P

Answer 2

Do you have multiple choice answers? Also it's not really possible to find it without a perimeter.

Answer 3

no its not multiple choice. its optimization

Answer 4

Chuk, let me help.

Answer 5

i think i'm supposed to take the derivative but i always get confused when the variable is in the denominator

Answer 6

2(512/y)+2y=P
-> multiply everything by y.
-> the result is a quadratic (P is a constan)

Answer 7

@ChukRock , you don't need derivatives, although your prof probably wants them.

Answer 8

so i should get 1024+2y^2=p

Answer 9

btw, with the resulting quadratic, you'll want to set the derivative to 0, and do dP/dy, because P is a function f y.

Answer 10

*=py

Answer 11

Actually, that makes it messier. Don't multiply, sorry.

Answer 12

i think taking the deriv got me (2048/y^2)+2=p

Answer 13

Can we redo the problem, and just get rid of the y? I'll teach you how to set it up.

Answer 14

well i kinda want to learn the way of taking the derivative because i'm sure the prof will want to see work

Answer 15

I know, I mean resetting up the problem so you only need 2 variables.

Answer 16

well at the point i got it to wouldnt i replace the p with zero and its only one variable?

Answer 17

Well, you actally did the your setup wrong. should've been py, not p.

Answer 18

Let me show you what I would do.

Answer 19

ok

Answer 20

x(p/2-x)=512
Minimize P.

Answer 21

i've never seen this way

Answer 22

Because, I got rid of the y.

Answer 23

P=2x+2y ->P/2=x+y -> P/2-x=y
xy=512

Answer 24

The right substitution makes things easier. :)

Answer 25

so (xp/2)-x^2=512? maybe it makes it easier for you but now i'm even more lost

Answer 26

find dP/dx and set it to zero. Use implicit differention as well as the product rule. If you haven't learned those, wel, I'll think of another way.

Answer 27

the way i learned was to substitute either x=(512/y) or y=(512/x) into the objective function

Answer 28

i'm just trying to figure out how to do it THAT way

Answer 29

That's all good, but getting that 1/y is harder to solve when you find the derivative.

Answer 30

thank you anyway

Answer 31

in any case, you'd wind up with implicity differentiation. I just realized that your method is actually faster.

Answer 32

But, that doesn't keep you from having to use the product rule and implicit differentiaion.

Answer 33

the answer (obviously) is 16sqrt2

Answer 34

@inkyvoyd i figured it out the way to do it with derrivs and you dont need chain or implicit diff\[2(512/y)+2y=P\]\[(1024/y)+2y=P\]\[-1024/y ^{2}+2=P\] at this point P becomes zero\[2=1024/y ^{2}\]\[2y ^{2}=1024\]\[y ^{2}=512\]\[y=16\sqrt{2}\]

Answer 35

That's half correct. You forgot to divide by y on the right hand side of the equation, but the answer still comes out.