A home owner wants to fence in his yard of 600 sq. feet. Three sides are to be constructed with chain link fence at a cost of 3.25 per linear feet. The fourth side is to be constructed with a wood stockade fence at a cost of $5.45 per linear foot.
A) what is a complete model for the cost of a building the fence in terms of one single input?
Minimize: c= 3.25(2w+l)+5.45(l)
Constraint: A=lw=600
To create a complete model for the cost of building the fence, we need to define the variables and equations that represent the problem.
Let's start by assigning a variable to the length of the three sides that will be constructed with a chain link fence. Let's call this variable "x". Since we know the dimensions of the yard, we can calculate the length of the fourth side (which will be constructed with a wood stockade fence) by subtracting the total length of the three sides from the perimeter of the yard.
The perimeter of the yard is twice the sum of the length and width. Let's assume the length of the yard is "L" and the width is "W". Therefore, the perimeter can be represented as:
Perimeter = 2(L + W)
Since the yard is rectangular, we know that the area is equal to the length multiplied by the width:
Area = L * W
We are given that the area of the yard is 600 sq. feet. So we can write the equation:
L * W = 600
Now, to calculate the length of the fourth side, we can substitute "W" in the perimeter equation with 600 / L:
Perimeter = 2(L + (600 / L))
To find the cost of the chain link fence, we multiply the length of each side (x) by the cost per linear foot, which is $3.25:
Cost of chain link fence = 3 * 3.25 * x
Similarly, the cost of the wood stockade fence can be calculated by multiplying the length of the fourth side by the cost per linear foot, which is $5.45:
Cost of wood stockade fence = 5.45 * (Perimeter - 3 * x)
Finally, the complete model for the cost of building the fence in terms of a single input "x" can be written as:
Cost = Cost of chain link fence + Cost of wood stockade fence
Cost = (3 * 3.25 * x) + (5.45 * (2(L + (600 / L)) - 3x))
In this model, you can substitute any value for "x", which represents the length of the three sides constructed with chain link fence, and calculate the corresponding cost of building the fence.