An offshore oil well is 4 kilometers off the coast. The refinery is 5 kilometers down the coast (see figure). Laying pipe in the ocean is twice as expensive as on land. What path should the pipe follow in order to minimize the cost?
I'm sorry!
It's a 30-60-90 triangle with 4 on the sqr of 3 side, x on the 1 side and 5-x extending past the triangle
so, you seem to have the diagram down. Now just define the cost function and minimize it.
What do you get? Is your 30-60-90 triangle you final answer, or some intermediate stage?
What’s the answer
To minimize the cost of laying the pipe, we need to find the path that requires the least amount of pipe to be laid in the ocean (the more expensive option).
Here's how we can approach this problem step-by-step:
1. Let's start by drawing a diagram to visualize the situation. Draw a line to represent the coast, and mark the offshore oil well 4 kilometers away from the coast, and the refinery 5 kilometers down the coast.
2. Now, let's consider the possible paths the pipe can follow to connect the oil well and the refinery. Since laying pipe in the ocean is twice as expensive as on land, we want to minimize the distance of pipe laid in the ocean.
3. One possible option is a direct path from the offshore oil well to the refinery, traveling a straight line from the oil well to the coast, and then following the coastline to the refinery. This path would involve laying 4 kilometers of pipe in the ocean and 5 kilometers of pipe on land. So, in total, it would require 4 kilometers of expensive ocean pipe and 5 kilometers of cheaper land pipe, resulting in a total cost of 4 * 2 + 5 = 13 units.
4. Another option is a path that minimizes the length of pipe laid in the ocean. In this case, the pipe would travel along the coastline until it is directly opposite the refinery (at a point 5 kilometers from the coast). It would then go straight from there to the refinery. This path would involve laying 5 kilometers of pipe in the ocean and 5 kilometers of pipe on land. So, in total, it would require 5 kilometers of expensive ocean pipe and 5 kilometers of cheaper land pipe, resulting in a total cost of 5 * 2 + 5 = 15 units.
5. Comparing the two options, we can see that the first option with a direct path from the oil well to the refinery along the coast results in a lower cost of 13 units, compared to 15 units for the second option. Therefore, the path that minimizes the cost is the one that follows a straight line from the oil well to the coast, and then follows the coastline to the refinery.
By analyzing the cost implications of different path options, we can determine the most cost-effective solution.