Question

Find the dimensions of a rectangular area of 256 square meters that has the least perimeter.

Answer 1

\[xy=256\]\[2x+2y=P(x,y)\]

Answer 2

I got 16 and 16, but I don't know if there are any other dimensions.

Answer 3

I got 16 , 16 as well. I think we can just trust each other on this. Thank you!

Answer 4

\[xy=256\]\[2(x+y)=P\]\[y=\frac{256}{x}\]Substitution\[2(x+\frac{256}{x})=P\]\[2x+\frac{512}{x}=P\]\[P'=2-\frac{512}{x^2}\]Find critical point\[0=2-\frac{512}{x^2}\]\[2x^2=512\]\[x^2=256\]\[x=±16\]Solve for y\[xy=256, x=16\]\[16y=256\]\[y=16\]A 16x16 rectangular area will have the least perimeter