Question

A dairy farmer plans to enclose a rectangular pasture adjacent to a river. To provide enough grass for the herd, the pasture must contain 304,200 square meters. No fencing is required along the river. What dimensions will use the least amount of fencing

Answer 1

Answer:

390 m (perpendicular to river) x 780 m (parallel to river)

Step-by-step explanation:

Let y be the length of the side parallel to the river, and let x be the length of the sides perpendicular to the river.

The total area and length of fence required are given by:

$A=304,200=xy\\y=\frac{304,200}{x} \\L=2x+y$

Rewriting the length of fence as a function of only x:

$L=2x+\frac{304,200}{x}$

The value of x for which the derivate of L(x) is zero is the length of x that uses the least amount of fencing:

$L=2x+\frac{304,200}{x}\\\frac{dL}{dx}=2- \frac{304,200}{x^2}=0\\x^2=152,100\\x=390$

If x = 390 m, then:

$y=\frac{304,200}{390}\\y=780$

The dimensions that will use the least amount of fencing are 390 m x 780 m