A rancher has 800 feet of fencing to enclose two adjacent rectangular corrals.
Write the total area A of the corrals as a function of x.
What dimensions produce a maximum enclosed area?
x = __ft
y = __ft
Thanks
Well, since there are two adjacent rectangular corrals, let's call one of them Corral 1 and the other Corral 2.
Now, let's assume that Corral 1 has a length of x feet and a width of y feet. Since the two corrals are adjacent, Corral 2 will also have a length of x feet, but its width will be 800 - y feet (since the total fencing length is 800 and we are already using y feet of it for Corral 1).
To find the area of Corral 1, we can simply multiply its length x by its width y, so its area is A1 = x*y. Similarly, the area of Corral 2 is A2 = x*(800-y).
Since we want to find the total area A of the corrals, we can simply add the areas of Corral 1 and Corral 2:
A = A1 + A2
= x*y + x*(800-y)
= xy + 800x - xy
= 800x
So, the total area A of the corrals can be represented as a function of x as A(x) = 800x.
To find the dimensions that produce a maximum enclosed area, we need to find the value of x that maximizes the function A(x).
However, since A(x) = 800x is a linear function, it does not have a maximum value. It will keep increasing as x increases without bound.
Therefore, any positive value of x will produce the maximum enclosed area, since the function keeps increasing.
So, we can say that x = any positive value and y = 800 - y will give us the dimensions that produce a maximum enclosed area.
To write the total area A of the corrals as a function of x, we need to determine the dimensions of each rectangular corral.
Let's assume that one corral has a width of x feet. Since there are two adjacent corrals, the other corral would also have a width of x feet.
Now, we need to determine the lengths of the corrals. Let's say the length of one corral is y feet. Since the total length of all the sides of the corrals equals the fencing available, we can calculate the total length as follows:
Length of one corral + Length of adjacent corral + Width of both corrals = Total fencing
y + y + x + x = 800
Simplifying the equation:
2y + 2x = 800
y + x = 400
Since we have two variables, this equation alone is not enough to determine the specific dimensions of the corrals.
To find the dimensions that produce the maximum enclosed area, we can express the area A as a function of x and then find the maximum value of that function.
The area of one corral can be calculated by multiplying its length and width:
Area of one corral = x * y
Since there are two corrals, the total area A would be:
A = 2 * (x * y) = 2xy
To determine the maximum value of A, we need to find the critical points of the function. Taking the derivative of A with respect to x and setting it equal to zero gives us:
dA/dx = 2y = 0
Since y is not a function of x, this implies that y = 0.
However, having a width of zero does not make sense in this context, so we can discard this critical point and conclude that there is no maximum area enclosed in this scenario.
Therefore, we cannot determine the specific dimensions that produce a maximum enclosed area with the given information.
You probably have a field with 2 long sides and 3 shorter sides?
long side --- y
short side --- x
2y + 3x = 800
y = (800-3x)/2 = 400 - (3/2)x
area = xy
= x(400 - 3x/2) = 400x - (3/2)x^2
again, just like in your other post, find the vertex of this parabola
using the method that you learned.