A rancher wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle (see the figure below). He has 1440 feet of fencing available to complete the job. What is the largest possible total area of the four pens?
We cannot see the figure. You cannot copy and paste here.
To find the largest possible total area of the four pens, we need to determine the dimensions of the rectangular area that will maximize the total area.
Let's denote the width and length of the rectangular area as w and l, respectively.
Since the fencing is parallel to one side of the rectangle, we need to have two widths and two lengths of the rectangle for the four pens.
We need to find the maximum area, which can be done by maximizing the product of the width and length. Therefore, our goal is to maximize the function A = w * l.
Now, let's consider the total amount of fencing available.
The perimeter of the rectangular area is given as 1440 feet, which means the sum of all four sides of the rectangle should be 1440 feet.
We can write this as the equation: 2w + 2l = 1440.
Now, let's solve this equation and express either w or l in terms of the other variable.
2w + 2l = 1440
Dividing both sides by 2, we get:
w + l = 720
l = 720 - w
Now, substitute the expression for l in terms of w into the formula for the total area A = w * l.
A = w * (720 - w) = 720w - w^2
Now, we have our area function A in terms of w.
To maximize A, we need to find the vertex of the graph of the quadratic function A = 720w - w^2.
The vertex of a quadratic function occurs at w = -b/2a, where the equation is in the form Ax^2 + Bx + C. In our case, a = -1, b = 720, and c = 0.
w = -b/2a = -720 / 2(-1) = 720/2 = 360
So, the width of the rectangular area that maximizes the total area is 360 feet.
Substitute this value back into the equation l = 720 - w to find the length of the rectangular area.
l = 720 - 360 = 360 feet
Now, we can calculate the area A by multiplying the width and length of the rectangular area:
A = w * l = 360 * 360 = 129,600 square feet
Therefore, the largest possible total area of the four pens is 129,600 square feet.