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.
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.
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.
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
a. To find the expression for the perimeter P in terms of x and y, let's start by labeling the dimensions of the plot as shown in the diagram. Let's call the length of the plot x and the width of the entire plot (including the additional fenced area) y.
The perimeter of the rectangular plot can be calculated by adding the lengths of all sides:
P = x + y + x + y
Since one additional fence is parallel to the sides of length x, it will contribute an additional y to the total perimeter. Similarly, the other additional fence parallel to the sides of length y will contribute an extra x to the perimeter.
Therefore, the expression for the perimeter P in terms of x and y is:
P = 2x + 2y + y + x
Simplifying further, we get:
P = 3x + 3y
b. We are given that the area of the plot is 1500 m². The area of a rectangle is given by the product of its length and width, so we have:
xy = 1500
From this equation, we can rewrite y in terms of x:
y = 1500/x
Substituting this value of y into the expression for P from part a, we get:
P = 3x + 3(1500/x)
Simplifying further, we have:
P = 3x + 4500/x
c. To graph P as a function of x, we can plot P on the y-axis and x on the x-axis. Choose a range of values for x and calculate the corresponding values of P using the equation P = 3x + 4500/x. Plot these points on a graph and connect them to create the graph of P as a function of x.
d. To determine the value of x that leads to the smallest value of P, we need to find the minimum point on the graph of P as a function of x. We can do this by finding the x-value where the slope of the graph is zero.
Taking the derivative of P with respect to x:
dP/dx = 3 - 4500/x^2
Setting the derivative equal to zero and solving for x:
3 - 4500/x^2 = 0
4500/x^2 = 3
4500 = 3x^2
x^2 = 4500/3
x^2 = 1500
x = √1500
x ≈ 38.73
Given that the plot is divided into two equal areas, we can find y by substituting the value of x into the equation xy = 1500:
y = 1500/x
y ≈ 1500/38.73
y ≈ 38.73
Therefore, the dimensions of the plot that result in using the least amount of fencing are approximately x ≈ 38.73 meters and y ≈ 38.73 meters.
e. To determine the smallest value of P, we substitute the values of x and y into the expression for P:
P = 3x + 3y
P = 3(38.73) + 3(38.73)
P ≈ 232.38 + 116.19
P ≈ 348.57 meters
Therefore, the smallest value of P is approximately 348.57 meters.