The owner wants the field to contain 125,000 square meters. No fencing is required along the river. What dimensions (in meters) will use the least amount of fencing?

Question

A campground owner plans to enclose a rectangular field adjacent to a river. The owner wants the field to contain 125,000 square meters. No fencing is required along the river. What dimensions (in meters) will use the least amount of fencing?

x=

 y=

Answer 1

Let the sides be L and W

LW = 125,000

W = 125,000/L

P= perimeter = 2L + 2W= 2L + 125,000/L

take the derivative and set equal to zero, to solve for the L that minimizes perimeter

P' = 2 - 250,00/L^2 = 0

multiply by L^2 to get

2L^2 = 500,000

L^2 = 250,000

L = sqr(250,000) = 500 m

w= 125,000/500 = 250 m

Answer 2

Let's assume x = length of fencing and y = width of fencing. The perimeter of the fence can be represented by P = x + 2y since one of the lengths will not have fencing along side of it.

We also know the area A = x * y = 125,000

We want to minimize the fence perimeter which means we want to determine P' = 0 so we must solve for an equation of P first. Let's use the area formula to do that:

x = 125,000/y

Substituting into the perimeter equation, we get:

P = 125,000y^-1 + 2y

Now let's solve for P':

P' = -1(125,000y^-2) + 2 = 0

-125,000/y² = -2

-125,000 = -2y²

62,500 = y²

y = 250 m

Since x = 125,000/y, we can solve for x:

x = 125,000/250 = 500 m