An electric utility is required to run a cable from a transformer station on the shore of a lake to an island. The island is 6 km from the shore and the station is 12 km down the shoreline from a point opposite the island. It costs $4000/km to run the cable on land and $6000/km underwater. Find the path the cable should take for a minimum cost of installation.
Did you make a diagram?
Mine has I as the island, O as the point on shore opposite the island, and T as the transformer station.
So IO = 6 km and OT = 12 km
Let P be any point between O and T,
let OP = x, then PT = 12-x
Cost = 6000(IP) + 4000(12-x)
IP^2 = x^2 + 6^2
IP = (x^2 + 36)^(1/2)
Cost = 6000(x^2 + 36)^(1/2) + 48000 - 4000x
d(Cost)/dx = 3000(x^2+36)^(-1/2)(2x) - 4000
= 0 for a max/min of Cost
2x(3000)/√(x^2 + 36) = 4000
3x/√(x^2 + 36) = 2
square and cross multiply
9x^2 = 4x^ + 144
5x^2 = 144
x= √144/5 = 5.3667
Form your word conclusion
say it goes 12-x km down the shoreline, then heads offshore straight for the island. The resulting path length is 12-x + the hypotenuse sqrt(36 + x^2)
cost = (12 -x)(4000) + [(36+x^2)^.5]6000
0 =dc/dx = -4000 + .5 [ (36+x^2)^-.5 ] (2x)(6000)
which is
4000 = 6000x/sqrt(36+x^2)
sqrt(36+x^2) = (3/2)x
36+x^2 = (9/4)x^2
36 = (5/4)x^2
x^2 = 28.8
x = 5.37 km
12-x = 6.63 miles on shore then head under water
Well, running a cable from the shore to an island is not as straightforward as telling a funny joke. But I'll certainly give it a shot!
To find the path with the minimum cost of installation, we need to consider both the land and underwater options. Let's break it down step by step:
1. From the transformer station to the nearest point on the shore opposite the island, the distance is 12 km. We'll run the cable on land for this segment, costing $4000/km. So, the cost for this part is 12 km * $4000/km = $48,000.
2. Now, we need to get the cable from the point on the shore opposite the island to the island itself, which is 6 km away. Since it's an underwater stretch, it will cost $6000/km. So, the cost for this part is 6 km * $6000/km = $36,000.
3. Finally, we need to consider the distance from the transformer station to the island. We can calculate this using the Pythagorean theorem since we have a right-angle triangle formed by the shore, the point opposite the island, and the island. The distance is the hypotenuse, √(12^2 + 6^2) = √180 = 13.42 km. Since it's mostly underwater, the cost is 13.42 km * $6000/km = $80,520.
Adding up the costs for all three segments, we get $48,000 + $36,000 + $80,520 = $164,520.
So, the path that minimizes the cost of installation is to run the cable on the shore for the first 12 km, then underwater for 6 km, and finally underwater for 13.42 km. Voila, we've solved this electric (and aquatic) puzzle!
To find the path the cable should take for a minimum cost of installation, we can use the concept of optimization and find the minimum cost path.
Let's break down the problem into smaller steps:
Step 1: Identify the possible paths for the cable to run.
In this case, we have two options: running the cable along the shoreline or underwater across the lake.
Step 2: Calculate the cost of running the cable along the shoreline.
The distance from the station on the shore to a point opposite the island is 12 km. So, the cost of running the cable along the shoreline would be:
Cost_shoreline = 4000 $/km * 12 km
Step 3: Calculate the cost of running the cable underwater.
The distance between the opposite point on the shore and the island is 6 km. So, the cost of running the cable underwater would be:
Cost_underwater = 6000 $/km * 6 km
Step 4: Calculate the total cost for each path option.
If we choose the shoreline path, the total cost would be the cost of running the cable along the shoreline:
Total_cost_shoreline = Cost_shoreline
If we choose the underwater path, the total cost would be the cost of running the cable underwater plus the cost of running the cable along the shoreline from the station to the opposite point on the shore:
Total_cost_underwater = Cost_underwater + Cost_shoreline
Step 5: Compare the total costs of each path option and choose the one with the minimum cost.
Compare Total_cost_shoreline and Total_cost_underwater.
If Total_cost_shoreline is less than Total_cost_underwater, then the cable should be run along the shoreline.
Otherwise, if Total_cost_underwater is less than Total_cost_shoreline, then the cable should be run underwater.
In this case, we need to calculate the specific costs and compare them to make the final decision.
To find the path that minimizes the cost of cable installation, we can use a method known as the Shortest Path Problem. In this case, the problem involves finding the minimum cost path from the transformer station to the island, considering both land and underwater installation costs.
One approach to solve this problem is by using Dijkstra's algorithm. Here's how you can proceed:
1. Create a graph to represent the problem. Each point along the shoreline, including the transformer station and the point opposite the island, should be represented as a vertex. The edges connecting these vertices will represent the cable installation paths.
2. Assign a weight to each edge based on the installation cost. Since it costs $4000 per kilometer on land and $6000 per kilometer underwater, you should calculate the length of each edge and multiply it by the corresponding installation cost.
3. Apply Dijkstra's algorithm to find the shortest path from the transformer station to the island. This algorithm will calculate the minimum cost path by considering the weights assigned to each edge.
4. Once the algorithm reaches the island vertex, you will have the minimum cost path with the corresponding installation cost.
By following these steps, you should be able to determine the path the cable should take for the minimum cost of installation.