A cattle rancher wants to enclose a rectangular area and then divide it into five pens with fencing parallel to one side of the rectangle (see the figure below). There are 560 feet of fencing available to complete the job. What is the largest possible total area of the five pens?
So there are two long sides and 6 short sides. Maximum area will be when the fencing is divided equally between the two dimensions. In this case, that would be 280 ft each. So the area will be 140 by 46.67 = 6533.33 ft^2
2 L + 6 W = 560 ... L + 3 W = 280
L = 280 - 3 W
a = L * W = 280 W - 3 W^2
amax is on the axis of symmetry
Wmax = -280 / (2 * -3)
Probable solution expected by the instructor:
let the length of the whole compound be y
let the width of each pen be x
given: 2y + 6x = 560
y = 280 - 3x
area = xy
= x(280-3x) = 280x - 3x^2
d(area)/dx = 280 - 6x
= 0 for a max of area
6x = 280
x = 280/6 = 140/3
y = 280 - 3(140/3) = 140
max area = (140)(140/3) = 19600/3
the max area is 19600/3 ft^2
To find the largest possible total area of the five pens, we need to determine the dimensions of the rectangular area that will maximize the area while using 560 feet of fencing.
Let's assume the length of the rectangular area is L and the width is W.
Since there are five pens, we need four partitions to divide the rectangular area into five sections. These partitions will run parallel to the length (L) of the area.
Since each partition will have the same length as the width (W), we will use 4W feet of fencing for the partitions.
The remaining fencing, which is available to enclose the perimeter of the rectangular area and the partitions, is given by:
Fencing available = Perimeter of rectangle + Fencing for partitions
560 = 2L + 3W + 4W
560 = 2L + 7W
Rearranging the equation, we get:
2L = 560 - 7W
L = (560 - 7W) / 2
Now, we need to express the total area in terms of W. The area of each pen is given by:
Area of a pen = L * W
So, the total area of the five pens is:
Total area = 5(L * W) = 5LW
Substituting the value of L, we have:
Total area = 5((560 - 7W) / 2) * W
To maximize the total area, we need to find the value of W that maximizes the expression.
One way to do this is by graphing the equation and finding the maximum point. However, since we're explaining how to find the solution, let's use calculus to find the critical points.
Taking the derivative of the total area with respect to W:
d(Total area)/dW = 0
Now, solve for W to find the critical points.
5(560 - 7W) / 2 = 0
560 - 7W = 0
7W = 560
W = 80
Now, substitute this value of W back into the equation for L to find the corresponding value:
L = (560 - 7(80)) / 2
L = 200
So, the dimensions of the rectangular area that will maximize the total area are L = 200 and W = 80.
Finally, to find the largest possible total area, substitute these values into the equation for the total area:
Total area = 5 * (200 * 80)
Total area = 80,000 square feet
Therefore, the largest possible total area of the five pens is 80,000 square feet.