A farmer has 25 yards of fencing to make a pig pen. He is going to use the side of the barn as one of the sides of the fence, so he only needs to fence 3 sides. What should be the dimensions of the fence in order to maximize the area?
Area = Length * Breadth
Now we know that the perimeter in this case is of three sides i.e.:
2(Breadth) + Length = 25
So if the breadth is x, the length is 25-2x
So, area
= x(25-2x)
= 25x - 2x^2
Taking f(x) = 25x - 2x^2
f'(x) = 25 - 4x
f''(x) = -4
Equating f'(x) to zero, we see that (25/4) gives the required value of x to maximize the area.
Therefore,
Breadth = 25/4 yards
Length = 25 - 2x = 25/2 yards
To maximize the area, the farmer should aim for a rectangular fence, as it is the shape that encloses the largest area for a given perimeter.
Let's denote the length of one side of the fence as x yards, and the width (or the other side) as y yards. We know that the sum of the lengths of all three sides is 25 yards. Since one side is already taken care of by the barn, the remaining two sides would add up to 25 - x yards.
So, based on this information, we can set up the equation:
x + 2y = 25 - x
Now, rearrange the equation:
2x + 2y = 25
To find the dimensions that maximize the area, we need an equation in terms of only one variable. In this case, we can use the equation for the area of a rectangle: Area = length x width.
The area, A, is represented as:
A = x * y
We substitute the value of x from the first equation into the equation for the area:
A = (25 - 2y) * y
To find the maximum area, we can take the derivative of A with respect to y, set it equal to zero, and solve for y:
dA/dy = 25 - 4y = 0
Solving this equation, we find:
y = 25/4 = 6.25
Now we can substitute this value of y back into the first equation to find the corresponding value of x:
x + 2(6.25) = 25 - x
x = 12.5
Therefore, to maximize the area, the farmer should build a rectangular fence with dimensions of 12.5 yards by 6.25 yards.