suppose that you are a power company who must provide power to a lighthouse on an island 1 mile off shore. the nearest pole to run the line from is 2 miles up the shore from the line straight out to the lighthouse. if the line costs $1000/mile on land and $1500/mile on water, find the path that minimizes the total cost. should state how far from the pole the line leaves the shore to go to the lighthouse. shot that it is a minimum and find the cost.
land distance: x
water distance: √(1^2+(2-x)^2)
Now determine the cost c(x) and find its minimum.
so would the water distance just be 2-x?
15 x 100 =1500 km 360 x 12=720 + 3600=4320 km
To find the path that minimizes the total cost, we can use the concept of optimization and consider the cost function as the objective we want to minimize. Let's break down the problem into smaller steps:
Step 1: Define the variables:
Let's denote the distance from the pole to the point where the line leaves the shore as x (in miles). We need to find the value of x that minimizes the total cost.
Step 2: Determine the cost function:
The cost of running the line on the land section of the path is $1000 per mile, and on the water section, it is $1500 per mile.
So, the cost function can be represented as:
Cost(x) = cost of land section + cost of water section
Step 3: Calculate the cost of the land section:
The distance from the pole to the point where the line leaves the shore is 2 miles. Hence, the cost of the land section is given by:
Cost of land section = $1000 * (2 - x)
Step 4: Calculate the cost of the water section:
The distance from the point where the line leaves the shore to the lighthouse is the hypotenuse of a right-angled triangle with one side measuring x miles and the other side measuring 1 mile.
By applying the Pythagorean theorem, the distance to the lighthouse is given by:
Distance to lighthouse = √(x^2 + 1^2)
The cost of the water section is then:
Cost of water section = $1500 * Distance to lighthouse
Step 5: Calculate the total cost:
To get the total cost, we need to add the costs of the land section and the water section:
Total Cost = Cost of land section + Cost of water section
Substitute the calculated values from step 3 and step 4 into the equation above.
Step 6: Find the x-value that minimizes the total cost:
To find the value of x that minimizes the total cost, we need to differentiate the total cost function with respect to x, set it to zero, and solve for x.
Step 7: Check if the critical point is a minimum:
Once you find the critical point(s) by solving the equation from step 6, check if the second derivative is positive at that point. If the second derivative is positive, the critical point is a minimum.
Step 8: Calculate the cost for the minimum path:
Substitute the x-value obtained in step 7 into the total cost equation to find the minimum cost.
By following these steps, you can find the path that minimizes the total cost of running the power line to the lighthouse and determine the distance from the pole where the line leaves the shore, as well as the associated cost.