A farmer wants to make a rectangular pen for his chickens. He has 1200 m of fencing. He will use one side of his barn for the pen. What dimensions will give the maximum area? What is the maximum area?
L + 2 W = 1200
A = L * W ... substituting ... a = (1200 - 2 W) * W = 1200 W - 2 W^2
the max area is on the axis of symmetry of the parabola
... Wmax = -b / 2 a = -1200 / (2 * -2) = 300
substitute back to find length and area
To find the dimensions that give the maximum area, we need to set up an equation and then differentiate it to find the critical points. Let's assume the length of the pen is 'L' and the width is 'W'.
Given the information, we know that the total length of fencing is 1200 m, and one side of the barn is already covered, meaning we only need to fence three sides. We can set up the equation for the total length of fencing:
Total Length of Fencing = Length of one long side + 2 * Width + Length of one end
1200 = L + 2W
Now, we can express the length L in terms of W:
L = 1200 - 2W
To find the area of the pen, we need to multiply the length and width:
Area = L * W
= (1200 - 2W) * W
Now we have an equation for the area in terms of the width W. To find the dimensions that give the maximum area, we need to differentiate this equation with respect to W and set it to zero:
d(Area)/dW = 0
Let's differentiate the equation:
d(Area)/dW = (1200 - 2W) + 0 * W - 2
= 1200 - 2W - 2
Setting this derivative equal to zero:
1200 - 2W - 2 = 0
Simplifying the equation:
-2W = -1200 + 2
W = 601
Now that we have the width, we can substitute it back into the equation for L:
L = 1200 - 2W
L = 1200 - 2(601)
L = 1200 - 1202
L = -2
Since negative values don't make sense in this context, we discard the negative value of L. Therefore, the width is 601m, and the length is 0m.
So, the dimensions that give the maximum area are a width of 601m and no length (just the side of the barn).
To find the maximum area, substitute the width value into the area equation:
Area = (1200 - 2W) * W
Area = (1200 - 2(601)) * 601
Area = (1200 - 1202) * 601
Area = -2 * 601
Area = -1202
Again, negative values don't make sense in this context, so we discard the negative value. Therefore, the maximum area is 1202 square meters.
To find the dimensions of the rectangular pen that will give the maximum area, we need to use the derivative to find the critical points.
Let's assume the length of the pen is L and the width is W. We know that the perimeter of the pen (excluding the side of the barn) is 1200 m.
Since we are using one side of the barn as a fence, the perimeter equation becomes:
2L + W = 1200
We need to express W in terms of L, so we can substitute it into the area equation.
From the perimeter equation, we get:
W = 1200 - 2L
Now, let's express the area (A) of the pen in terms of L:
A = L * W
A = L * (1200 - 2L)
A = 1200L - 2L²
To find the maximum area, we need to find the derivative of the area equation and set it equal to zero:
dA/dL = 1200 - 4L
Set this equation equal to zero and solve for L:
1200 - 4L = 0
4L = 1200
L = 300
Now that we have the value of L, we can substitute it back into the perimeter equation to find the width W:
2L + W = 1200
2(300) + W = 1200
600 + W = 1200
W = 600
Therefore, the dimensions that will give the maximum area are L = 300 m and W = 600 m.
To find the maximum area, substitute the values of L and W into the area equation:
A = L * W
A = 300 * 600
A = 180,000 square meters
So, the maximum area for the rectangular pen is 180,000 square meters.