Optimization problem.

[babby]

So I'm struggling a little with this optimization problem. I'm kind of winging it and teaching this to myself and I was wondering if someone could point out the mistakes in my work?
Find the dimensions of a rectangle with area 343000 m² whose perimeter is as small as possible. (Give your answers in increasing order, to the nearest meter.)
So I figured that A=LxW (length times width) and P=2L+2W. So I put W in terms of L, or W=343000/L. Taking the derivative of my new perimeter function, I got p'(l)=2-(686000/l²) and then I found the zero of the graph, which is equal to approx. 586 meters, which in turn is equal to L², so I took the square root of L and I figured that would be one of my values for my dimensions but it is incorrect. Where did I go wrong?

 

[mmm4444bot]

Technical Notes: It's hard to read math steps when they are compressed into a single paragraph. Please, in the future, type each equation on its own line. Also, do not interchange upper- and lower-case letters; symbols p and P (or l and L) do not have the same meaning.


Is this what you got for P in terms of L?

P = 2L + 686000/L

 

[mmm4444bot]

I found the zero of the [perimeter function], which is equal to approx. 586 meters, which in turn is equal to L²

I figured that would be one of my values for my dimensions but it is incorrect.

Yes, P'(586) is zero (rounded), but the zero of the derivative function is not the same as L². What do you intend, when you say, "which in turn is equal to L^2"?

Who told you that 586 meters is not one of the dimensions?

 

[soroban]

Hello, babby!

A strange problem . . . Who wrote it?

Find the dimensions of a rectangle with area 343000 m² whose perimeter is as small as possible.
(Give your answers in increasing order, to the nearest meter.)

That area happens to be \(\displaystyle 70^3\)
. . which is a "nice" number for a volume problem
. . but not for an area problem.

The optimal shape turns out to be a square,
. . so the side of the square is \(\displaystyle 70\sqrt{70}\)


Couldn't they have chosen a "nicer" number, like 576 or 8100?
. . Or even a smaller one, like 27 or 43?

 

[HallsofIvy]

So I'm struggling a little with this optimization problem. I'm kind of winging it and teaching this to myself and I was wondering if someone could point out the mistakes in my work?
Find the dimensions of a rectangle with area 343000 m² whose perimeter is as small as possible. (Give your answers in increasing order, to the nearest meter.)
So I figured that A=LxW (length times width) and P=2L+2W. So I put W in terms of L, or W=343000/L. Taking the derivative of my new perimeter function, I got p'(l)=2-(686000/l²) and then I found the zero of the graph, which is equal to approx. 586 meters, which in turn is equal to L², so I took the square root of L and I figured that would be one of my values for my dimensions but it is incorrect. Where did I go wrong?

You had already defined "L" as being the length of one side Why in the world would you take the square root and expect that to be a length?

 

[babby]

Is this what you got for P in terms of L?

P = 2L + 686000/L

Yes.

A strange problem . . . Who wrote it?

It is a homework problem from a Calculus textbook by James Stewart.

Yes, P'(586) is zero (rounded), but the zero of the derivative function is not the same as L². What do you intend, when you say, "which in turn is equal to L^2"?

Who told you that 586 meters is not one of the dimensions?

Now that you say that, I realize that I was making it a lot harder than it actually was. I was following a similar example where it appeared that they did something similar to what I did.

Thank you all for the help, I figured out my mistake.