A farmer wishes to build a fence for 6 adjacent rectangular pens. If there is 600 feet of fencing available, what are the dimensions of each pen that maximizes total pen area?
The image looks like this:
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also:
If the interior fencing is $3.00 per foot and the perimeter is $5.00 per foot, what are the pen dimensions that minimize cost?
assume each pen encloses 200 square feet of area in the 2nd part sorry!
part 2:
if each pen has width x and height y
xy=200
6x+8y=600
you have no room to vary the dimensions. So, I assume the 600 feet of fencing does not apply to part 2. Accordingly,
xy=200
cost c = 5(3x+4y)+3(3x+4y)
= 24x+32y
= 24x+32(200/x)
dc/dx = 24 - 6400/x^2
= 8(3x^2-800)/x^2
dc/dx = 0 when x = 20√(2/3)
so, the pens are 20√(2/3) by 10√(3/2)
25'x25'
To find the dimensions that maximize the total pen area, we can start by assigning variables to the dimensions of the rectangle. Let's say the length of each rectangle is "L" and the width is "W."
Since there are six adjacent rectangular pens, we can calculate the total amount of fencing required by adding up the lengths of the rectangular pens. In this case, the total length of the fencing required is 6L. Given that there are 600 feet of fencing available, we can set up the equation:
6L = 600
Dividing both sides of the equation by 6, we find that the length of each pen is L = 100 feet.
Now, we can calculate the total width of the fencing required by considering the width of each pen and the length of the pen. Since each pen is rectangular, the width will be W. So, the total width of the fencing required is 6W.
We also know that the perimeter of each pen is given by the formula: Perimeter = 2L + 2W. Since each pen has the same dimensions, we can calculate the perimeter of each pen and multiply it by 6 to find the total perimeter required. In this case, the total perimeter required is 6(2L + 2W).
Given that there is a total of 600 feet of fencing, we can set up the equation:
6(2L + 2W) = 600
Dividing both sides of the equation by 6, we find:
2L + 2W = 100
Now, we can use the equation 2L + 2W = 100 to express one variable in terms of the other. Let's solve for L:
2L = 100 - 2W
L = 50 - W/2
Next, we need to find an expression for the total area of the six rectangular pens. The area of each pen is given by the formula: Area = Length × Width. Since each pen has the same dimensions, we can calculate the total area by multiplying the area of each pen by 6:
Total Area = 6(L × W)
= 6(50 - W/2) × W
= 300W - 6W^2
To find the dimensions that maximize the total pen area, we need to maximize the Total Area function. We can do this by finding the vertex of the quadratic function 300W - 6W^2. The vertex will give us the maximum value of the function.
The x-coordinate of the vertex is given by: x = -b/2a
In this case, a = -6 and b = 300. Plugging these values into the formula, we find:
W = -300 / 2(-6)
W = -300 / -12
W = 25
Now that we have the value of W, we can substitute it into the equation to find the value of L:
L = 50 - W/2
L = 50 - 25/2
L = 50 - 12.5
L = 37.5
Therefore, the dimensions that maximize the total pen area are: Length (L) = 37.5 feet and Width (W) = 25 feet.
Moving on to the second question about minimizing the cost:
To find the dimensions that minimize cost, we need to consider the cost of the interior fencing and the cost of the perimeter fencing.
Let's assign the length of each pen as "L" and the width as "W" as before.
The cost of the interior fencing is $3.00 per foot. Since there are six pens, the total cost of the interior fencing is given by: 6(L + W) × $3.00.
The cost of the perimeter fencing is $5.00 per foot. The perimeter for each pen is given by: Perimeter = 2L + 2W. Multiply this by 6 to get the total perimeter required and multiply it by $5.00 to get the cost: 6(2L + 2W) × $5.00.
The total cost is the sum of the cost of the interior fencing and the cost of the perimeter fencing:
Total Cost = 6(L + W) × $3.00 + 6(2L + 2W) × $5.00
To find the dimensions that minimize the cost, we need to minimize this cost function.
Simplifying the equation gives us:
Total Cost = 18L + 18W + 60L + 60W
= 78L + 78W
Since we want to express one variable in terms of the other, let's solve for W:
W = (Total Cost - 78L) / 78
Now, we can substitute this expression for W into the equation for the total area (Total Area = L × W) to find the dimensions that minimize the cost.
Total Area = L × [(Total Cost - 78L) / 78]
Differentiating the Total Area equation with respect to L and setting it equal to zero will give us the minimum value of Total Area. By taking the derivative and solving for L, we can find the value of L that minimizes the Total Area. Once we have L, we can substitute it back into the expression for W.
However, since the specific values for the Total Cost and the cost per foot are not provided, it is not possible to determine the exact dimensions that minimize cost without further information.