A farmer with 1000m of fencing wants to enclose a rectangula

A farmer with 1000m of fencing wants to enclose a rectangular plot that borders on a river. If the farmer does not need a fence along the river side (the length), what are the dimensions of the largest plot. What is the area of the largest plot? (for your answer enter only the numbers without units)

Dimensions: length

width

the largest area is

Solution

Perimeter of rectangle = 2(length + width)

Length of fencing = 1000m

1000 = 2(width) + length

Let length = x

Width = y

1000 = x+2y

x = 1000-2y

A = xy

A = (1000-2y) * y

A = 1000 y - 2y²

For maximum area

dA/dy = 1000 - 4y

1000 - 4y = 0

-4y = -1000

y = 1000/4

y = 250

x = 1000 - 2y = 1000 - 2*250 = 1000 - 500 = 500

Length = 500

Width = 250

The largest area = 500 *250 = 125000