A farmer wishes to fence off a rectangular pasture along the bank of a river. The area of the pasture is 3200 $yd^2$. There is no fencing needed along the river bank. Find the dimensions of the pasture that'll require the least amount of fencing. $$\\$$
I'm not sure how to begin this problem.
I know the area is $lw$ = 3200.
Please help.
Since the pasture is a rectangle, the Area= $xy$= 3200.
The minimum fencing required is f($x$,$y$) = $x$+2$y$.$$\\$$ $fx$→ 1 = λy
$\implies$ $\frac{1}{y}$ = λ
$fy$→ 2 = λx
$\implies$ $\frac{2}{x}$ = λ
Set them equal and cross multiply: $\frac{1}{y}$=$\frac{2}{x}$
$\implies$ 2$y$=$x$ $$\\$$ Remember that the Area: $xy$= 3200 and we just found what $x$ equals.
$xy$= 3200
= (2$y$)$y$ = 3200
= 2$y^2$ = 3200
= $y^2$ = 1600
= $y$ = ±40
$y$ = 40 because you can't have a negative yard. $$\\$$
Remember that 2$y$=$x$ and we just found what $y$ equals.
2$y$=$x$
= 2(40) = $x$
= $x$ = 80 $$\\$$ So your dimension are 80 x 40 yards.