Optimization: A farmer has 2400 ft of fencing and wants to fence off a rectangular field...

A farmer has 2400 ft of fencing and wants to fence off a rectangular field that boards a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area?
Learn how to solve optimization problems for your Calculus 1 class.
This problem is taken from the Single Variable Calculus textbook by James Stewart, 9th ed. Get the notes here: / my-calculus-1-81985618
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[@kingbeauregard]

If i could choose a shape for maximum area, i think i would choose a semicircle. How would i prove that semicircles are optimal?

[@bra1nwave172]

I'm not an expert but I think you would have to use calculus of variations that would be similar to something called the isoperimetric problem that MathVerse made a video about.

[@ostorjulien2562]

I managed to prove it thanks to Euler-Lagrange equations with equality constraints. You have to write the problem in terms of the curve described by the fence, represented by a function f. The aera under the curve y=f(x) must be maximal while the length of the fence has to be equal to 2400.

[@kingbeauregard]

@@ostorjulien2562 That's fantastic, thank you!

[@ajinkya3.14]

If you have the perimeter, a square will always give you the largest area

[@tux1968]

In the given problem, where you have 2400ft of fence for 3 sides of the field, the largest square possible would be x=800, and y=800, with an area of 640,000. But that would have been less area than the non-square answer of x=600, and y=1200, with an area of 720,000.

[@nobodyspecial7895]

But the very video you're commenting on proves that you're wrong lol.

[@ajinkya3.14]

​@@tux1968​Indeed, but with just three sides, you don't have the complete perimeter. My statement holds true only when the complete perimeter is given, you can even try it out for yourself.
For example: if the perimeter is 10 and we take all the possible sides for a rectangle and calculate their area, the sides that are equal (length=5 and breadth=5) will give you the largest area. In case the perimeter is 11 units, the sides should be 5.5 and 5.5 units.
In short, even perimeters have same integer sides for largest area while odd perimeters have same decimal sides.
Hope this clears the mist :)

[@SannidhyaSamanta]

Sir, what happened to the whiteboard??

[@Abhay0505]

Epic 😊

[@bingcheng8419]

A silly question: Can cows swim?

[@CornishMiner]

Not boards, borders.

[@rolflangius1119]

You have to prove that using a shorter length cannot result in a larger area