A dog kennel with four pens is to be constructed. The pens will be surrounded by rectangular fence that costs $13 per meter. The rectangle is partioned into four pens of equal size with three partitions made of fence that costs $5 per meter. Each pen measures x meters wide by y meters long.
a) What is the total cost of the fence?
b)Suppose each pen must have 9 square meters of area. What should x and y be to minimize the cost of the fence?
(a) c = 13(4x+y+4x+y) + 5(3y)
= 112x+41y
(c) xy=9, so y=9/x, and
c(x) = 112x + 369/x
now find x where dc/dx = 0
a) To calculate the total cost of the fence, we need to consider two types of fences: the rectangular fence that surrounds the entire kennel and the partitions within the kennel.
The rectangular fence:
The perimeter of the rectangular fence is equal to the sum of all four sides. Since we know the width (x) and length (y) of each pen, the perimeter of the rectangular fence is given by:
P = 2(x + y)
The cost of the rectangular fence per meter is $13, so the total cost of the rectangular fence is:
C_rectangular = 13P = 13 * 2(x + y)
The partitions within the kennel:
Each pen is partitioned by three fences that cost $5 per meter. The total length of the fences used for partitions in each pen is:
L_partition = 2x + 2y
Since there are four pens, the total length of the partition fences is:
L_total_partitions = 4(L_partition) = 8x + 8y
The cost of the partition fences per meter is $5, so the total cost of the partition fences is:
C_partitions = 5(L_total_partitions) = 5(8x + 8y)
The total cost of the fence is the sum of the cost of the rectangular fence and the cost of the partition fences:
Total cost = C_rectangular + C_partitions
Total cost = 13 * 2(x + y) + 5(8x + 8y)
Total cost = 26(x + y) + 40x + 40y
Total cost = 66x + 66y
b) To minimize the cost of the fence, we need to minimize the value of the total cost equation. Since both x and y are in the equation, we need to find the values of x and y that will minimize the equation.
We are given that each pen must have an area of 9 square meters. The area of each pen is given by:
A = x * y = 9
To minimize the cost equation, we can solve for y in terms of x using the area equation:
y = 9 / x
Substituting this value of y into the total cost equation:
Total cost = 66x + 66(9 / x)
Total cost = 66x + 594 / x
To minimize the cost, we can take the derivative of the total cost equation with respect to x and set it equal to zero:
d(Total cost) / dx = 66 - 594 / x^2 = 0
Simplifying this equation:
66x^2 - 594 = 0
x^2 = 594 / 66
x^2 = 9
x = sqrt(9)
x = 3
Substituting this value of x back into the area equation:
y = 9 / x
y = 9 / 3
y = 3
Therefore, to minimize the cost of the fence, the width (x) and length (y) of each pen should be 3 meters.
To find the solution to this problem, let's break it down step by step:
a) What is the total cost of the fence?
To calculate the total cost of the fence, we need to consider two parts: the cost of the outer rectangular fence and the cost of the partitions inside.
1. Cost of the outer rectangular fence:
The outer rectangular fence surrounds all four pens. Since there are four sides to the rectangular fence and no partitions along these sides, we can calculate the total cost by multiplying the perimeter by the cost per meter, which is $13.
The perimeter of a rectangle is given by the formula 2(x + y), where x is the width and y is the length.
Cost of the outer rectangular fence = perimeter * cost per meter
= 2(x + y) * 13
= 26(x + y)
2. Cost of the partitions:
Inside the rectangle, there are three partitions splitting it into four equal-sized pens. These partitions have a cost per meter of $5. Since there are three partitions separating the four pens, we can calculate the total cost by multiplying the perimeter of each partition by the cost per meter and then multiplying by 3.
Cost of partitions = 3 * (perimeter of partitions * cost per meter)
= 3 * [(2x + 2y) * 5]
= 30(x + y)
Overall, the total cost of the fence is the sum of the cost of the outer rectangular fence and the cost of the partitions:
Total cost of the fence = Cost of the outer rectangular fence + Cost of partitions
= 26(x + y) + 30(x + y)
= 56(x + y)
Therefore, the total cost of the fence is given by the expression 56(x + y).
b) To minimize the cost of the fence while each pen has an area of 9 square meters, we need to find the values of x and y that would minimize the total cost.
Given that each pen has an area of xy = 9, we can solve for one variable in terms of the other, i.e., x in terms of y or y in terms of x.
xy = 9
=> x = 9/y
Substituting this expression for x in the total cost equation:
Total cost of the fence = 56(x + y)
= 56((9/y) + y)
= 56(9 + y^2)/y
To minimize the total cost, we need to find the value of y that minimizes this expression. We can do this by taking the derivative with respect to y and setting it equal to zero, then solving for y. However, this requires a more advanced mathematical approach.
So, in conclusion, the total cost of the fence is given by the expression 56(x + y). To minimize the cost while each pen has an area of 9 square meters, we need to find the values of x and y that would minimize this expression, which requires further mathematical analysis.