An offshore well is 2 miles off the coast. The refinery is 4 miles down the coast. If a laying pipe in the ocean is twice as expensive as on land, what path the pipe should follow in order to minimize the coast.
Is the answer 2.1? Can you some list their steps?
Well, well, well, let me pipe in with a pun-tastic answer for you! If laying a pipe in the ocean is twice as expensive as on land, we want to minimize the cost. So, instead of taking a straight path right from the offshore well to the refinery, we need to consider an alternate route.
Here's a step-by-step guide to minimize the cost:
1. Start by drawing a straight line from the offshore well to the refinery. A direct path, let's call it "A".
2. Now, draw another line from the offshore well that is perpendicular to the coast until it intersects with the straight line connecting the well and the refinery. Let's call this new point of intersection "B".
3. Calculate the distance from the offshore well to point "B" along the land. Let's call it "x".
4. Calculate the distance from point "B" to the refinery along the coastline. This distance would be "2 - x" miles.
5. Now, let's find the total cost for this zigzagging path. Since laying a pipe in the ocean is twice as expensive as on land, we'll need to multiply the land distance "x" by 1 (for the cost of laying land pipes), and the ocean distance "2 - x" by 2 (for the cost of laying ocean pipes).
6. Add the two costs together to find the total cost.
7. Repeat steps 2-6, gradually changing the position of point "B" along the straight line connecting the offshore well and the refinery.
8. Keep moving point "B" along the line until you find the position that gives you the minimum total cost.
So, if the answer you got is 2.1, it might be the optimal position for point "B". Keep in mind that to be completely accurate, you'll need to calculate the cost for each possible position of point "B" along the straight line and choose the one with the minimum cost.
To determine the path that minimizes the cost, we need to consider the costs of laying the pipe in the ocean versus on land. According to the given information, laying the pipe in the ocean is twice as expensive as on land.
Let's analyze the different scenarios:
1. If the pipe goes straight from the offshore well to the refinery along the coast:
- Distance traveled in the ocean: 2 miles
- Distance traveled on land: 4 miles
- Total cost: (2 miles * 2) + (4 miles * 1) = 4 + 4 = 8
2. If the pipe goes straight from the offshore well to the refinery through land:
- Distance traveled in the ocean: 0 miles
- Distance traveled on land: 6 miles
- Total cost: (0 miles * 2) + (6 miles * 1) = 0 + 6 = 6
As we can see, scenario 2, where the pipe goes straight from the offshore well to the refinery through land, has a lower total cost of 6 compared to scenario 1, which has a total cost of 8.
Therefore, the path that minimizes the cost is for the pipe to go straight from the offshore well to the refinery through land. The optimum path is not mentioned as being 2.1 miles; it is 6 miles along the coast.
To determine the path the pipe should follow to minimize the cost, we can consider the different costs associated with laying the pipe in the ocean and on land. Let's break down the problem step by step:
1. First, let's visualize the situation. Draw a diagram with the offshore well (point A), the refinery (point B), and the coast line in between.
2. We are given that the offshore well is 2 miles off the coast, and the refinery is 4 miles down the coast. Therefore, the straight-line distance between the well (A) and the refinery (B) is 4 miles.
3. Let's assume that laying the pipe in the ocean costs $X per mile, and laying the pipe on land costs $Y per mile. Given that laying the pipe in the ocean is twice as expensive as on land, we can set up the following equation: X = 2Y. This means that the cost of laying the pipe in the ocean is twice the cost of laying it on land.
4. Now, to minimize the cost, we need to consider whether it is more cost-effective to lay the pipe in the ocean or on land.
5. Let's consider two possible paths for the pipe:
a) Straight from the offshore well to the refinery, passing through the ocean.
b) Along the coast from the offshore well to the refinery, avoiding the ocean.
6. Calculate the cost for each path:
a) The straight path through the ocean will be a distance of 4 miles. Since laying the pipe in the ocean costs X per mile, the total cost for this path will be 4 * X.
b) The path along the coast will be a distance of 2 miles offshore (from well A to the coast) plus a distance of 4 miles along the coast (from the coast to the refinery). Since laying the pipe on land costs Y per mile, the total cost for this path will be (2 + 4) * Y = 6 * Y.
7. Since we are trying to minimize the cost, we need to compare the two costs calculated in step 6. As we know that X = 2Y, we can substitute X with 2Y in the equation: 4 * X = 6 * Y.
8. Simplifying the equation, we have: 4 * 2Y = 6 * Y. This reduces to 8Y = 6Y.
9. If we subtract 6Y from both sides, we get 8Y - 6Y = 6Y - 6Y, which simplifies to 2Y = 0.
10. Solving for Y, we find that Y = 0.
11. Since Y is equal to 0, it means that laying the pipe on land is free, and laying the pipe in the ocean is twice as expensive, which is 0 * 2 = 0.
12. Therefore, the minimum cost is 0, and the pipe should follow the path along the coast from the offshore well to the refinery to minimize the cost.
So, the answer is not 2.1; rather, the pipe should follow the path along the coast (avoiding the ocean) to minimize the cost, resulting in a total cost of 0.
Make a sketch, mark the well as W and the point on the shore closest to W as A, label the refinery as R
Place P somewhere between A and R.
They will lay the pipeline from R to P along the shore, then directly through the water to W
let AP=x , the PR = 4-x, and AW=4
by Pythagoras:
WP = √(9-x^2) or (9-x^2(^(1/2)
cost = 1(4-x) + 2(9-x^2)^(1/2) , the actual money units don't matter as long as they are in the ratio 1 : 2
d(cost)/dx = -1 + (1/2)(9 - x^2)^(-1/2) (-2x)
= 0 for a min of cost
1 = x/√(9-x^2)
x = √(9-x^2)
square both sides
x^2 = 9-x^2
2x^2 = 9
x^2 = 4.5
x = √4.5 = appr 2.12
So they should aim for a point P which is 2.12 miles from A or 1.88 miles from R