A farmer has 2,400 feet of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. Write the function that will produce the largest area if x is the short side of the rectangle.
2x+y = 2400
so, y = 2400-2x
a = xy = x(2400-2x) = 2400x-2x^2
That is just a parabola, with vertex at x=600
So, the field is 600 x 1200
As usual, these problems achieve maximum area when the perimeter is divided equally among lengths and widths.
To find the largest area of the rectangular field, we need to express the area in terms of one variable, x, and then maximize that function. Let's suppose the length of the rectangular field is denoted as L.
To start, we need to determine the perimeter of the rectangular field. The perimeter is equal to the sum of all four sides, which consists of two lengths (L) and two widths (x). According to the problem, we have a total of 2,400 feet of fencing available. Therefore, we can express the perimeter equation as:
2L + 2x = 2400
Next, we need to rewrite this equation in terms of L, so that we have the formula for the area in just the variable x. Solving the equation for L, we get:
2L = 2400 - 2x
L = 1200 - x
The area of the rectangular field is given by multiplying the length (L) and the width (x):
A = L * x = (1200 - x) * x
Finally, we write the function that produces the largest area (A) in terms of x:
f(x) = (1200 - x) * x
With this function, we can now find the maximum value of A by finding the critical points and using the First Derivative Test or by graphing the function and locating the highest point on the graph.
To find the largest area, we need to optimize the function for the given constraints. The area of a rectangle is given by the formula:
Area = length × width
In this case, the width will be the unknown, while the length is given as the distance along the river, which requires no fence. Let's assume the length is "L" and the width is "x" (short side of the rectangle).
Now, the perimeter (P) of the rectangle is given as 2x (width) + L (length), and it is also given that the total fencing available is 2,400 feet. Therefore, we have the equation:
P = 2x + L = 2400
Since we are looking to maximize the area, we can express the length (L) in terms of x, using the equation above:
L = 2400 - 2x
Now, substitute the value of L into the area formula:
Area = x × (2400 - 2x)
Therefore, the function that represents the largest possible area can be expressed as:
f(x) = x × (2400 - 2x)
We can now use calculus to determine the optimal value of x that maximizes the area.