a farmer has a 1500 feet of fencing in his barn.he wishes to enclose a rectangler pen,subdivided in two regions by a section of the 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.what is the domain of the function.?
To find the area enclosed by the pen as a function of its width, we need to understand the geometry of the problem.
Let's call the width of the rectangle x. Since the pen is subdivided by a section of the fence down the middle, we can divide x into two equal parts. Therefore, each half of the pen's width will be x/2.
Now, for each half of the pen, we have two sides parallel to the length (L) of the rectangle and one side parallel to the width (x/2). Additionally, we have two sections of the fence of length x/2 each (since we divide the pen down the middle). The remaining two sections of the fence, each of length L, are used for the two length sides of the rectangle.
To calculate the area enclosed by the pen, we need to find the total length of the two length sides (L) and the total length of the two width sides (x/2). These lengths should add up to the total length of the fence, which is given as 1500 feet.
Total length of the length sides = 2L
Total length of the width sides = 2(x/2) = x
We can write the equation: 2L + x = 1500
Next, we need to express L as a function of x. Solving the equation for L, we get:
2L = 1500 - x
L = (1500 - x) / 2
Finally, we can express the area (A) enclosed by the pen as a function of its width (x) by multiplying the length and width:
A = L * x = [(1500 - x) / 2] * x = (1500x - x^2) / 2
Now, let's consider the domain of the function. In this case, the width (x) represents a physical measurement, so it cannot be negative. Therefore, the domain of the function is x ≥ 0, meaning that the width must be greater than or equal to zero.
To summarize:
- The area enclosed by the pen is given by the function A(x) = (1500x - x^2) / 2.
- The domain of the function is x ≥ 0.