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
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
Just noticed that both Damon and I answered a very similar question in the first of the Related Questions below
http://www.jiskha.com/display.cgi?id=1268848081
I suggest you print it out , and change the numbers, and units.
To find the point at which the line should begin to angle in order to yield the minimum total cost, we need to analyze the cost of running the cable at different angles and find the minimum cost.
Let's assume that the point at which the line begins to angle is point C. The length between points A and C is given as 7 miles.
To determine the cost, we need to consider two segments: one running along the shoreline from A to C and the other underwater from C to B.
Let's represent the distance from A to C as x (in miles). This means the distance from C to B would be (6 - x) miles.
The cost of running the cable on land is $400 per mile, and the cost of running it underwater is $500 per mile.
The cost of the land segment, AC, can be calculated as 400 * x.
The cost of the underwater segment, CB, can be calculated as 500 * (6 - x).
The total cost, TC, is the sum of these two costs:
TC = 400x + 500(6 - x).
To find the minimum cost, we need to minimize the total cost function TC.
To minimize this function, we can take its derivative with respect to x and set it equal to 0. Then solve for x.
d(TC)/dx = 400 - 500 = 0
400x - 500(6 - x) = 0
400x - 500(6) + 500x = 0
400x + 500x - 3000 = 0
900x = 3000
x = 3000/900
x = 3.33 (rounded to two decimal places)
Therefore, the point at which the line should begin to angle is approximately 3.33 miles from A.
To find the point at which the line should start angling in order to yield the minimum total cost, we need to consider the costs of running the cable on land and underwater, and determine the point that minimizes the total cost.
Let's denote the point at which the line starts angling as point C, and the length between points A and C as x. We are given that the length between A and C is 7 miles, and the total length from A to B (including the submerged portion) is 6 miles.
First, let's calculate the length of the underwater portion, which is simply the total length (6 miles) minus the length of the land portion (x miles):
Underwater portion length = 6 miles - x miles
Next, we can calculate the cost of running the cable on land, which is $400 per mile, and the cost of running the cable underwater, which is $500 per mile:
Cost of running cable on land = $400/mile * x miles
Cost of running cable underwater = $500/mile * (6 miles - x miles)
The total cost is the sum of the costs on land and underwater:
Total Cost = Cost of running cable on land + Cost of running cable underwater
Total Cost = $400/mile * x miles + $500/mile * (6 miles - x miles)
To find the minimum total cost, we need to minimize this cost function. We can do this by finding the value of x that minimizes the cost function using calculus.
Derivative of the cost function with respect to x:
d(Total Cost)/dx = $400 - $500
Set the derivative equal to zero and solve for x:
$400 - $500 = 0
$400 = $500
We can see that there is no solution to this equation, which means that the cost function does not have a minimum value. This occurs because the cost of running the cable underwater is greater than the cost of running it on land.
Therefore, we cannot determine the point at which the line should begin angling to yield the minimum total cost in this scenario.