given the area of a rectangle, find the smallest possible rectangle (calculus optimization)

[@Checkmate_Central]

There's another problem like this one with an interesting solution, where you want to build a fence of the greatest area and smallest perimeter, but using a lake as one of the sides.

[@adrianyaguar7666]

Yes :) it will be rectangle with sides x and 2x, not a square :)

[@GirishManjunathMusic]

Before watching the video: xy = 1000 square units x = 1000y↑(-1) Minimise perimeter = 2(x + y): Differentiating 2(1000y↑(-1) + y) 2(1 - 1000/y²) = 2(1 + 10√10/y)(1 - 10√10/y) Setting equal to zero to find critical numbers: 1 + 10√10/y = 0 or 1 - 10√10y = 0 As this is a geometric problem, y cannot be negative, so we will go with option 2: 1 = 10√10/y → y = 10√10 units x = 1000/y, which makes x also 10√10 units. The perimeter is therefore minimised at 2(x + y) = 40√10 units.

[@opfis6882]

3:57 Y has to be equal to 1/1000

[@aleelopee_]

Doing this with a triangle instead of a rectangle should be an interesting problem!

[@Toninhophi19]

Wow, I love it

[@kaiudall2583]

So to minimize the perimeter of a rectangle given an area, it will be a square. Very cool

[@littleretroship6403]

Ok so here we go I’m starting to love optimization problems. xy=1000 y=1000/x A(x)= 2x + 2000/x A(x)= 2x + 2000x^(-1) A’(x)= 2 - 2000x^(-2) Set A’ to 0 0= 2 - 2000x^(-2) -2=-2000x^(-2) 2=2000x^(-2) 1/1000=1/x^2 1000=x^2 x=sqrt(1000) Perimeter is: 4*sqrt(1000) Oops mistake again

[@PLAYWORD]

I liked it, but I didn't understand anything hahaha

[@田村博志-z8y]

This is a basic application of AM-GM.

[@ethanyap8680]

Thought the same

[@jonthebob2159]

Am-gm bruh