Optimization
An offshore oil well is 2km off the coast. The refinery is 4 km down the coast. Laying a pipe in the ocean is twice as expensive as on land. What path should the pipe follow in order to minimize the cost?
To minimize the cost of laying the pipe, we need to find the path that requires the least amount of pipe on the expensive offshore section.
Let's break down the problem into smaller components:
1. Define the variables:
Let's assume the distance from the oil well to the refinery on land as x kilometers.
This means the distance from the oil well to the refinery through the ocean would be (4 - x) kilometers.
2. Calculate the cost:
The cost of laying pipe on land is denoted as C_land, and the cost of laying pipe in the ocean is denoted as C_ocean.
We are given that laying a pipe in the ocean is twice as expensive as on land.
Therefore, we can define the cost equation as follows:
Total Cost (C) = C_land * x + C_ocean * (4 - x)
3. Find the minimum cost:
To find the optimal path, we need to minimize the cost function.
Let's differentiate the cost function with respect to x and set it equal to zero to find the minimum value of x:
dC/dx = 0
4. Solve for the minimum cost path:
Taking the derivative of the cost function, we have:
dC/dx = C_land - C_ocean
Setting it equal to zero and solving for x:
C_land - C_ocean = 0
or
C_land = C_ocean
This equation tells us that the cost of laying the pipe on land is equal to the cost of laying the pipe in the ocean. Therefore, the pipe should follow a path where the distance covered on land is equal to the distance covered in the ocean.
In the given problem, the total distance from the oil well to the refinery is 4 km. Therefore, the optimal path would be where the pipe follows half the distance on land (2 km) and half the distance through the ocean (2 km).
So, to minimize the cost, the pipe should follow a path where the distance covered on land is 2 km, and the distance covered in the ocean is also 2 km.