A rancher wants to create a fence enclosing 1500 m2 of land. The plot will be divided into two equal areas by an additional fence parallel to two sides as shown in the diagram below.

Question

Find the dimensions of the plot which results in using the least amount of fencing.


a. Write an expression for the perimeter P of the fence in terms of x and y.

b. Use the fact that the area of the plot is 1500 m2 to rewrite P in terms of x alone.

c. Graph P as a function of x.

d. Use the graph to determine the value of x which leads to the smallest value for P. Use
this value of x to determine the value of y.

e. Use the graph to determine the smallest value of P.

Answer 1

If the field has length x and width y, then

xy = 1500
y = 1500/x

p = 2x+3y = 2x + 4500/x

See what you can do with that.

Answer 2

I assume this does not mean the perimeter but the total length of fencing.
I assume x is the length and there are 3 y fences

P = 2 x + 3 y

x y = 1500 so y = 1500/x
then
P = 2 x + 4500/x
x P = 2 x^2 + 4500
2 x^2 - xP = -4500
x^2 -xP/2 = -2250

x^2 -xP/2 + P^2/16 = -2250 + P^2/16
(x-P/4)^2 = P^2/16 -2250=(1/16)(P^2-36000)
vertex at
x = P/4 = 47.5
P = sqrt 36000 = 190 meters
so x = 47.5 meters
so y = 31.6 meters