A company wishes to run a utility cable from point A on
the shore to an installation at
point B on the island. The island is 6 miles from the shore. It
please explain it for me step by step without link it to a similar question: costs $400 per mile to run the cable on land and $500 per mile
underwater. Assume that the cable starts at A and runs along
the shoreline, then angles and runs underwater to the island.
Find the point at which the line should begin to angle in order
to yield the minimum total cost. the length between A AND C is 7
I assume that C is the point on shore closest to B.
So, if the pipeline leaves the shore at point P, between A and C, and the distance PC = x, the length p of the pipeline is
p(x) = (7-x) + √(x^2+36)
now we see that the cost c of the pipeline is
c(x) = 400(7-x) + 500√(x^2+36)
Now just find the minimum of c(x).
To find the point at which the line should begin to angle in order to yield the minimum total cost, let's break down the problem into steps:
Step 1: Identify the variables
- Let's call the point at which the line starts to angle as point C, where AC is the length between A and C.
- Let's also define x as the distance from C to B, where BC is the length between B and C.
Step 2: Find the total cost function
- The total cost is the sum of the cost for running the cable along the shoreline and the cost for running the cable underwater.
- The cost of running the cable along the shoreline is given as $400 per mile, so the cost for AC is 400 * AC.
- The cost of running the cable underwater is given as $500 per mile, so the cost for BC is 500 * BC.
- Therefore, the total cost can be represented as Cost = 400 * AC + 500 * BC.
Step 3: Express BC in terms of x
- We know that the total distance from A to B is 6 miles, and the sum of AC and BC should be 6 miles.
- So, BC can be expressed as 6 - AC - x.
Step 4: Simplify the total cost function
- Substituting BC with 6 - AC - x in the total cost function, we get Cost = 400 * AC + 500 * (6 - AC - x).
- Simplifying further, Cost = 400 * AC + 500 * (6 - AC) - 500 * x.
- Expanding, Cost = 500 * 6 - 100 * AC - 500 * x.
Step 5: Find the derivative of the cost function
- To find the point at which the total cost is minimized, we need to find the minimum point of the cost function. This can be done by finding the derivative of the cost function with respect to AC and setting it to zero.
- Taking the derivative of the cost function, we get dCost/dAC = -100 + 500 = 0.
- Solve the equation -100 + 500 = 0 for AC, we find AC = 5 miles.
Step 6: Find the value of x
- Now that we know AC = 5 miles, we can substitute it back into the equation BC = 6 - AC - x.
- BC = 6 - 5 - x, which simplifies to BC = 1 - x.
Step 7: Determine the optimal point C
- The problem asks us to find the point at which the line should begin to angle to yield the minimum total cost.
- Since AC is the distance between A and C, and we know that AC = 5 miles, the optimal point C is 5 miles from point A.
Therefore, the line should begin to angle at point C, which is 5 miles from point A, in order to yield the minimum total cost.