Given ensuing information, determine the least cost and the least cost mix of on-
and off-shore pipe.
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
Answer:
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
HOWEVER, I do not get how to get √(x^2+4) and √((3-x)^2 + 1)
Please explain!
To understand how to get √(x^2+4) and √((3-x)^2 + 1), we need to break it down step by step.
Let's start with √(x^2+4):
1. The expression inside the square root, x^2+4, represents the distance from the intake facility straight down to the shore.
2. The variable x represents the distance at which the pipe comes ashore from the point directly south of the intake facility.
3. To find the distance from the intake facility to the shoreline, we use the Pythagorean theorem.
- The vertical distance is 2 miles north of the shoreline.
- The horizontal distance is x miles.
- Therefore, the square of the hypotenuse (distance from the intake facility to the shoreline) is equal to (2^2 + x^2).
4. Taking the square root of (2^2 + x^2) gives us √(x^2+4) as the distance from the intake facility to the shore.
Now let's move on to √((3-x)^2 + 1):
1. The expression inside the square root, (3-x)^2 + 1, represents the distance from the water filtration plant straight up to the shoreline.
2. The variable x still represents the distance at which the pipe comes ashore from the point directly south of the intake facility.
3. To find the distance from the water filtration plant to the shoreline, we again use the Pythagorean theorem.
- The vertical distance is 1 mile south of the shoreline.
- The horizontal distance is (3-x) miles.
- Therefore, the square of the hypotenuse (distance from the water filtration plant to the shoreline) is equal to ((3-x)^2 + 1).
4. Taking the square root of ((3-x)^2 + 1) gives us √((3-x)^2 + 1) as the distance from the water filtration plant to the shore.
By plugging in these expressions into the cost function c(x) and finding where the derivative of c(x) is equal to 0, we can determine the least cost and the least cost mix of on- and off-shore pipe.
I hope this explanation helps clarify how to obtain √(x^2+4) and √((3-x)^2 + 1) in the given scenario!