A farmer has 1500 feet of fencing in his barn. He wishes to enclose a rectangular pen. Subdivided into two regions by a section of fence down the middle, parallel to one side of the rectangle. Express the area enclosed by the pen as a function of its width x.
So you have 2 long sections (x) and 3 short sections (y).
3x+2y = 1500
The area is A = xy = x(1500-3x)/2
or, 2x+3y = 1500
whichever way you define width
To express the area enclosed by the pen as a function of its width x, we need to consider the dimensions of the rectangular pen.
Let's assume the length of the rectangular pen is L, and the width is x. Since the pen is subdivided into two regions, we can divide it into two separate rectangles.
So, we have two rectangles with the same width x and lengths (L/2).
The total length of the fencing required for the rectangular pen is the sum of the perimeters of the two rectangles:
2L + x + x = 2L + 2x
But we are given that the farmer has 1500 feet of fencing, so:
2L + 2x = 1500
Now, let's solve this equation for L:
2L = 1500 - 2x
L = (1500 - 2x)/2
L = 750 - x
Now, to find the area enclosed by the pen, we multiply the length (L) by the width (x):
Area = L * x
Area = (750 - x) * x
Area = 750x - x^2
Therefore, the area enclosed by the pen as a function of its width x is given by the equation: Area = 750x - x^2.
To find the area of the pen, we first need to express its dimensions in terms of the given width, x. Let's assume that the length of the rectangular pen is L.
Since the fence is divided into two regions, we will have three sides to consider: the width (x), the length on one side of the divider (L), and the length on the other side of the divider (also L).
Now, let's calculate the amount of fencing used for each side:
1. Number of horizontal sides (width): Since there are two widths, the total length of fencing used for the widths would be 2x.
2. Number of vertical sides (lengths): There are two lengths (L) and a section of fence down the middle, which means there are three lengths of fencing. Each length is equal, so we have a total length of 3L.
Given that the total length of fencing provided is 1500 feet, we can create an equation to represent this:
2x + 3L = 1500
Now, let's solve for L in terms of x. Subtracting 2x from both sides, we have:
3L = 1500 - 2x
Dividing both sides by 3:
L = (1500 - 2x) / 3
To find the area of the rectangular pen, we use the formula A = length × width:
A = L × x
A = [(1500 - 2x) / 3] × x
Simplifying further, we can express the area (A) enclosed by the pen as a function of its width (x) using the equation:
A(x) = (1500x - 2x^2)/3
This equation gives the area of the pen for any given width value (x).