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?
length of side parallel to barn ---- y
length of each of the other two sides --- x
y + 2x = 25
y = 25-2x
area = xy = x(25-2x) = -2x^2 + 25x
d(area)/dx = -4x + 25 = 0 for a max of area
4x = 25
x = 25/4 , then y = 25 - 25/4 = 25/4
state your conclusion
or
from :
area = -2x^2 + 25x , a downwards parabola,
the x of the vertex = -b/(2a) = -25/-4 = 25/4 , as above
y = ...... , same result
To maximize the area of the pig pen, the fence should be in the shape of a rectangle. Since the farmer only needs to fence three sides and one side will be the barn, the pig pen will have two equal-length sides and one shorter side (the side opposite to the barn).
Let's assume the length of the two equal-length sides is 'x', and the length of the shorter side is 'y'. The perimeter of the pig pen is given by the equation:
Perimeter = 2x + y = 25
Now, we need to express the area of the pig pen in terms of 'x' and 'y'. The area of a rectangle is given by:
Area = length * width = x * y
To find the dimensions that maximize the area, we need a function in terms of a single variable. We can solve the perimeter equation for 'y' in terms of 'x' by rearranging the equation:
y = 25 - 2x
Substituting this value of 'y' in the area equation, we get:
Area = x * (25 - 2x)
To find the maximum area, we can take the derivative of the area function with respect to 'x' and set it equal to zero:
d(Area)/dx = (25 - 4x) = 0
Solving this equation, we find:
25 - 4x = 0
4x = 25
x = 25/4
x = 6.25
Now that we have the value of 'x', we can substitute it back into the perimeter equation to find the value of 'y':
2x + y = 25
2(6.25) + y = 25
12.5 + y = 25
y = 12.5
Therefore, the dimensions that maximize the area of the pig pen are:
Length of the two equal-length sides (x) = 6.25 yards
Length of the shorter side (y) = 12.5 yards