You have been hired by the Humane Society to construct six animal cages using 1400 feet of chain fence. Express the length and width using function notation. Include a graph with the area function with explanation of significance. Find the dimensions that maximize the total enclosed area and the relevant domain and range.
You need to specify more about the layout of the cages. Are they all stand-alone? Do they share sides in some way?
It's a 2x3 cage making the 6 cages
In that case, if each cage has width x and length y, the total fencing used is
9x+8y = 1400
and the area is
a = 6xy
But we know that y = (1400-9x)/8, so
a = 6x(1400-9x)/8 = 1050x - 27/4 x^2
That is just a parabola, with its vertex at x = 700/9
So, y = 700/8
You can finish up the other details.
Note that as usual in such problems, the fencing is divided equally among the lengths and the widths.
To find the length and width of the cages using function notation, let's assume the length of each cage is represented as "L" and the width is represented as "W".
Since we need to construct six cages, the total length of the chain fence needed will be six times the length of one cage (6L), and the total width will be six times the width of one cage (6W).
The total length of chain fence available is 1400 feet, so we can set up an equation to represent this:
6L + 6W = 1400
Divide both sides of the equation by 6 to isolate L and W:
L + W = 233.33
Now, let's express the length and width in function notation:
L = f(W) and W = f(L)
Next, let's create a graph to visualize the area function, which will help us find the dimensions that maximize the total enclosed area.
The area of a rectangle is given by the formula A = L * W.
In this case, the area of each cage (A) is given by the function:
A(W) = L(W) * W = f(W) * W
We can plot the graph with the width (W) as the x-axis and the area (A) as the y-axis.
Now, we need to determine the domain and range for this graph:
Domain (width, W): Since we are dealing with physical cages, it is reasonable to assume that the width cannot be negative or zero. Therefore, the domain for width (W) would be all positive real numbers: W > 0.
Range (area, A): The area cannot be negative, as it represents a physical measurement. So, the range for the area (A) would also be positive real numbers: A > 0.
To find the dimensions that maximize the total enclosed area, we need to find the maximum point on the graph. This can be done by finding the vertex of the parabolic shape that the graph represents.
To find the vertex, we can apply the principles of calculus by taking the derivative of the area function A(W) with respect to W, setting it equal to zero, and solving for W.
A'(W) = 0
Then, we substitute the value of W back into the equation for the length, L(W), to find the corresponding value of L.
This will give us the dimensions (length and width) that maximize the total enclosed area.
Please note that since we have not been given specific information about the shape or design of the cages, the above approach assumes rectangular cages with fixed dimensions. If there are any additional constraints or requirements for the cage design, the above method may need to be modified accordingly.