An oil well at O is in the ocean, 8 miles from T on a straight shoreline.
The oil has to go from O to R. the cost of laying pipe per mile is $90,000 underwater and $54,000 on land. the cheapest method of placing the pipe is to lay OP underwater and PR on land, where P is some point on RT. What is the amount of money that CAN BE Saved by using this method instead of going directly from O to R underwater?
using optimazatio can you please help me, I have no idea how to solve this problem.
There are many variations of this same question,.
Hint: let TP = x
then OP^2 = x^2 + 64
OP = (x^2 + 64)^(1/2)
I think you are missing the length of TR, we need that to find PR
Just for arguments sake, lets say TR = 12 miles
then PR = 12-x
cost = 90000(x^2+64)^(1/2) + 54000(12-x)
take the derivative, set that equal to zero and solve for x
Change the 12 miles to whatever you question has
To solve this problem using optimization, we can break it down into smaller steps:
Step 1: Define the variables:
Let the distance OP be represented by x (in miles).
The distance PR will then be (8 - x) miles.
Step 2: Set up the objective function:
The objective is to minimize the cost of laying the pipe.
The cost of laying OP underwater will be 90,000 * x dollars.
The cost of laying PR on land will be 54,000 * (8 - x) dollars.
The total cost can be represented as:
Total cost = 90,000x + 54,000(8 - x)
Step 3: Set up the constraints:
The distance OP must be less than or equal to 8 miles since it cannot exceed the distance between O and T along the shoreline.
So the constraint is:
x ≤ 8
Step 4: Maximize or minimize:
Since we want to minimize the cost, we will minimize the objective function.
Step 5: Solve the optimization problem:
To find the value of x that minimizes the total cost, we can take the derivative of the objective function, set it equal to zero, and solve for x.
d(Total cost)/dx = 90,000 - 54,000
Setting this derivative equal to zero, we get:
90,000 - 54,000 = 0
Therefore, 36,000 = 0
This equation has no real solutions, which means there is no local maximum or minimum for the objective function.
Step 6: Evaluate the endpoints:
To find the minimum cost, we need to evaluate the objective function at the endpoints of the feasible region.
Setting x = 0, we find:
Total cost = 90,000(0) + 54,000(8 - 0) = 432,000 dollars
Setting x = 8, we find:
Total cost = 90,000(8) + 54,000(8 - 8) = 720,000 dollars
Step 7: Conclusion:
The cheapest method of laying the pipe is by placing OP underwater and PR on land, where P is the point on RT that minimizes the cost.
The amount of money that can be saved by using this method instead of going directly from O to R underwater is the difference between the total cost using this method (432,000 dollars) and the total cost of going directly from O to R underwater (720,000 dollars):
Savings = 720,000 - 432,000 = 288,000 dollars.
To solve this problem using optimization, we will first determine the total cost of laying the pipe directly from O to R underwater. Then, we will find the cost of laying OP underwater and PR on land. Finally, we will calculate the difference between the two costs to determine the amount of money that can be saved.
Let's break down the problem step by step:
1. Calculate the cost of laying the pipe directly from O to R underwater:
- The distance from O to R underwater is the straight shoreline distance, which is 8 miles.
- The cost per mile of laying pipe underwater is $90,000.
- Multiply the distance by the cost per mile: 8 miles * $90,000 = $720,000.
2. Calculate the cost of laying OP underwater and PR on land:
- We know that P is some point on RT.
- To minimize the cost, we want to find the point P that divides RT in a way so that the cost of laying OP underwater and PR on land is minimized.
- We can use optimization techniques, such as finding the derivative of the cost function, to determine the value of P that minimizes the cost.
- However, since the problem does not provide any additional information or constraints, we cannot determine the exact value of P without more details.
3. Calculate the amount of money that can be saved:
- Once we have the cost of laying OP underwater and PR on land, we can calculate the difference between this cost and the cost of laying the pipe directly from O to R underwater.
- The difference will give us the amount of money that can be saved by using the cheaper method.
- Subtract the cost of laying OP underwater and PR on land from the cost of laying the pipe directly from O to R underwater: $720,000 - (Cost of OP underwater + Cost of PR on land).
Without additional information about the location or relative length of RT and OR, we cannot determine the exact value of P or the amount of money that can be saved.