A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river.

For largest rectangle
$x =2y$

 

$x + 2y = 2400$

$2y + 2y = 2400$

$4y = 2400$

$y = 600 ~ \text{ft}$
 

$x = 2(600) = 1200 ~ \text{ft}$

Total length of fence:
$x + 2y = 2400$

$x = 2400 - 2y$   ←   Eq. (1)
 

 

$A = xy$

$A = (2400 - 2y)y$

$A = 2400y - 2y^2$
 

For largest area of the field:
$\dfrac{dA}{dy} = 2400 - 4y = 0$

$2400 = 4y$

$y = 600 ~ \text{ft}$
 

From Eq. (1)
$x = 2400 - 2(600)$

$x = 1200 ~ \text{ft}$
 

Dimensions = 1200 ft wide along the river × 600 ft deep