A factory is located on one bank of a straight river that is 2000 m wide. On the opposite bank but 4500 m downstream is a power station from which the factory draws its electricity. Assume that it costs $3 per meter to lay an underwater cable and $1 per meter to lay an above ground cable.
How long should the cable above ground be in order to minimize the total cost?
To determine the optimal cable length, we need to consider the cost of laying an above ground cable versus an underwater cable for different lengths. Let's assume the length of the above ground cable is x meters.
The total cost (C) can be calculated as follows:
C = cost of above ground cable + cost of underwater cable
The cost of the above ground cable is $1 per meter, so it can be calculated as:
Cost of above ground cable = $1 * x
The cost of the underwater cable is $3 per meter, and it needs to cover the distance between the factory and the power station, which is the sum of the width of the river (2000 m) and the downstream distance (4500 m). So, the underwater cable cost can be calculated as:
Cost of underwater cable = $3 * (2000 + 4500)
Now, we can substitute these values into the total cost equation:
C = $1 * x + $3 * (2000 + 4500)
Simplifying:
C = $1 * x + $3 * 6500
C = x + $19500
To minimize the total cost (C), we need to find the value of x that minimizes it. Since the cost value will always increase with the length of the above ground cable, the minimum cost will occur when the above ground cable has zero length (x = 0).
Therefore, the above ground cable should be 0 meters long in order to minimize the total cost.
To solve this problem, we need to determine whether it is more cost-effective to lay the cable underwater or above ground. Let's consider the two scenarios and calculate the costs for each.
Scenario 1: Underwater Cable
Since the factory and the power station are located on opposite sides of the river, we need to lay the cable underwater to connect them. The width of the river is 2000 m, so the length of the underwater cable will be 2000 m.
Cost of laying an underwater cable = Length of cable x Cost per meter
= 2000 m x $3
= $6000
Scenario 2: Above Ground Cable
In this scenario, we need to find the length of cable that will minimize the total cost. To do this, we can divide the problem into two parts:
Part 1: Horizontal distance
The horizontal distance between the factory and the power station is given as 4500 m. Since we're not crossing the river, we can lay the cable above ground directly.
Cost of laying an above ground cable for the horizontal distance = Length of cable x Cost per meter
= 4500 m x $1
= $4500
Part 2: Vertical distance
To cross the river, we need to calculate the minimum length of cable required to connect the factory and the power station. This can be found using the Pythagorean theorem.
Length of the hypotenuse = √(length of horizontal distance^2 + width of the river^2)
= √(4500^2 + 2000^2)
= √(20250000 + 4000000)
= √24250000
≈ 4924.74 m
Cost of laying an above ground cable for the vertical distance = Length of cable x Cost per meter
= 4924.74 m x $1
= $4924.74
Total Cost of laying the above ground cable = Cost of part 1 + Cost of part 2
= $4500 + $4924.74
≈ $9424.74
Thus, the total cost of laying the above ground cable is approximately $9424.74.
By comparing the total costs for both scenarios, we can see that the cost of laying an underwater cable is $6000, while the cost of laying an above ground cable is approximately $9424.74. Therefore, it is more cost-effective to lay the cable underwater in this case.
If the cable comes out of the river x meters from the point directly opposite the factory, then the cost is
c = 3√(x^2+2000^2) + 1(4500-x)
so, find where dc/dx=0
The answer is 4500-x.