1 answer
(click or scroll down)
Explain Bot To find the maximum area that can be formed by a series of small pens next to a river with separate enclosures, we need to determine the dimensions of each pen that would result in the largest possible area.
Let's assume the width of each pen is denoted by "x" meters, and the length of each pen is denoted by "y" meters. The pens are placed adjacent to each other, so we have a series of x-meter-wide rectangular pens.
The total length of fencing available is given as 40 meters. Since there are two sides of each pen that do not require fencing (the river and the adjacent pen), and the remaining two sides of each pen require fencing, we can express the total length of fencing in terms of "x" and "y" as follows:
Length of fence = 2y + (n - 1)(2x), where "n" is the number of pens.
In this case, since the pens are placed adjacent to each other, "n" is equal to the number of pens.
Given that the total length of fencing is 40 meters, we can write the equation:
2y + (n - 1)(2x) = 40
However, we need to express "n" in terms of "x" to solve the problem. The number of pens, "n," can be obtained by dividing the total length of fencing by the width of each pen, "x":
n = 40 / x
Replacing "n" in the equation, we have:
2y + ((40 / x) - 1)(2x) = 40
Simplifying further, we have:
2y + (80 / x - 2)x = 40
2y + 80 - 2x = 40
2y - 2x = -40
Dividing both sides by 2, we get:
y - x = -20
Now, we need to express "y" in terms of "x" in order to find the maximum area. We can use the fact that the total area of all the pens is given by:
Area = nxy
Substituting "n" with the expression we derived earlier, we have:
Area = (40 / x)xy
Simplifying, we have:
Area = 40y
From the equation y - x = -20, we can derive y = -20 + x.
Substituting this into the area equation, we have:
Area = 40(-20 + x)
Area = -800 + 40x
To find the maximum area, we need to find the maximum value of this equation. Since the coefficient of "x" is positive, the maximum area will occur when x is maximum.
Given that the total length of fencing is 40 meters, the maximum value of "x" is 20 meters (half of the total length).
Substituting x = 20 into the area equation, we have:
Area = -800 + 40(20)
Area = -800 + 800
Area = 0
Therefore, the maximum area that can be formed by a series of small pens next to a river with separate enclosures is 0 square meters. This occurs when the width of each pen is 20 meters and the length of each pen is 0 meters.