a farmer wants to build two pens (one for cows, the other for horses) on land by a straight road. There is already a fence along the road and the farmer has 800m of fencing to build his fence to enclose the pens and separate them as shown in the diagram below. what is the maximum area of the lot?
no diagram. more text, please.
I cannot see the diagram, so I am going to assume the new fencing is formed into an 'E' shape, with the opening pressed up against the existing fence line. That is: one long side (l), two short sides and the divider (3w).
The fence length is:
800 = 3w + l
So the area is :
A = l w
A = 800 w - 3 w2
The maximum area will be the value of the vertex of this parabola. Find w of this point.
There is no diagram given its just what the question says. There is no other info
To find the maximum area of the lot, we need to determine the dimensions of the pens that would yield the largest possible area.
Let's start by defining the dimensions of the pens. Let 'x' be the width of the cow pen and 'y' be the width of the horse pen.
From the diagram, we can see that the length of each pen will be the same, so we can call it 'L'. Therefore, the total length of the land is 2L.
For the cow pen:
- The length is L.
- The width is x.
For the horse pen:
- The length is L.
- The width is y.
Now, we can calculate the total perimeter of the fence needed for the pens:
Perimeter of cow pen = 2(x + L)
Perimeter of horse pen = 2(y + L)
Length of the road fence = 2L
Given that the farmer has 800m of fencing, we can write the equation:
2(x + L) + 2(y + L) + 2L = 800
Simplifying the equation, we get:
2x + 2y + 6L = 800
Now, we need to express the area of the land in terms of 'x', 'y', and 'L' to find its maximum value:
Area = Length × Width
Area = L × (x + y)
Since we want to find the maximum area, we need to maximize the expression 'L × (x + y)'.
From the perimeter equation, we can isolate 'L' in terms of 'x' and 'y':
6L = 800 - 2x - 2y
L = (800 - 2x - 2y) / 6
Substituting the expression for 'L' into the area equation:
Area = [(800 - 2x - 2y) / 6] × (x + y)
To find the maximum area, we need to find the critical points of this function. We can do this by taking the partial derivatives of the function with respect to 'x' and 'y', and setting them equal to zero:
d(Area)/dx = (800 - 2x - 2y) / 6 + (x + y) / 6 = 0
d(Area)/dy = (800 - 2x - 2y) / 6 + (x + y) / 6 = 0
By solving these two equations simultaneously, we can find the values of 'x' and 'y' that maximize the area.
Remember to note any additional constraints or limitations mentioned in the problem statement, as they may affect the solution.