15. Given ensuing information, determine the least cost and the least cost mix of on-
and off-shore pipe.
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The intake facility is in the water 2 miles vertically north of the shoreline
The water filtration plant is on land 1 mile vertically south of the shoreline
The intake and the plant are horizontally 3 miles apart
The cost to lay pipe off-shore is $60,000 per mile and the cost to lay pipe on-
shore is $30,000 per mile
The shoreline runs east and west in a horizontal straight line
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If the pipe comes ashore x miles from the point directly south of the intake, the cost (in $K) is
c(x) = 60√(x^2+4) + 30√((3-x)^2 + 1)
dc/dx = 60x/√(x^2+4) - 30(3-x)/√((3-x)^2 + 1)
dc/dx=0 when x=1
So, we have √5 miles of offshore pipe and √5 miles of onshore pipe.
Cost = 90√5 $K = $201,246.12
can you tell me how you got √(x^2+4) and √((3-x)^2 + 1)
To determine the least cost and the least cost mix of on- and off-shore pipe, we need to calculate the cost for both options and compare them.
First, let's calculate the cost for the off-shore pipe. We know that the distance between the intake facility and the water filtration plant horizontally is 3 miles. Therefore, the cost for laying off-shore pipe would be:
Off-shore pipe cost = Distance * Cost per mile
Off-shore pipe cost = 3 miles * $60,000/mile
Off-shore pipe cost = $180,000
Secondly, let's calculate the cost for the on-shore pipe. We need to consider the vertical distance from the intake facility to the shoreline and from the shoreline to the water filtration plant.
Vertical distance from intake facility to shoreline = 2 miles
Vertical distance from shoreline to water filtration plant = 1 mile
Total vertical distance = 2 miles + 1 mile = 3 miles
On-shore pipe cost = Total vertical distance * Cost per mile
On-shore pipe cost = 3 miles * $30,000/mile
On-shore pipe cost = $90,000
Now we can compare the costs:
- Off-shore pipe cost: $180,000
- On-shore pipe cost: $90,000
Since the on-shore pipe cost is lower than the off-shore pipe cost, the least cost option is to use on-shore pipe only.
Therefore, the least cost mix of on- and off-shore pipe is to use on-shore pipe.