Calculus - Calculating Minimum and Maximum Values

- Calculus -
Calculating Minimum
and Maximum Values
Part III

Finally, we will use calculus for a real-world application.
The Ajax Water Company (point A), is located along a road (Line AB). You want to run water pipes so you can supply the town of Coolville (point C) with water. Laying pipe parallel to the road (Line AB) is easier and cheaper and costs 1 million dollars per mile. Laying pipe in the gray area is tougher and more expensive (rough terrain, heavily forested, etc) and costs 2.5 million dollars per mile.

The shortest route (but not the cheapest) is AC and is 5 miles but would cost 12.5 million dollars. Laying pipe along route ABC would cost 4 million dollars for 4 miles along Line AB and 7.5 million dollars for 3 miles along Line BC for a total of 11.5 million dollars. There is a route (AXC) that would cost even less than this. Here's how to calculate it.

We must set up an equation for the total cost.
The length of pipe laid parallel to the road AB equals distance AX. Line AX= (4 miles minus Line XB). Since the cost of laying pipe along AB is $1 million per mile, the cost of the pipe along Line AX equals
4 miles * ($1 million) - (XB miles *($1 million))
The length of pipe laid in the gray area is line XC. From the Pythagorean Theorem, we know that XC² = 3² + XB².
So, the cost of laying pipe along line XC equals
$2.5 million * SqRoot (XB² + 9).
For easier readability, numbers representing millions of dollars will be shown without the word "millions".
The equation for the total cost of this job is:

Cost Along Line AX Plus Cost Along Line XC
Which Equals:
4 * ($1) - XB*($1) Plus 2.5*(SqRoot (XB² + 9))
Multiplying and then adding up terms we get:
4 -XB + (6.25 XB² + 56.25)^½
Taking the derivative (by using the Chain Rule):
0 -1 + ½ * (6.25 XB² + 56.25)^-½ * (12.50 XB)

Moving the 1 to the "left side" of the equation:

12.50 XB
1= ---------------------------------------
2 * SqRt(6.25 XB² + 56.25)

Simplifying:

6.25 XB
1= ----------------------------------
SqRt(6.25 XB² + 56.25)

SqRt(6.25 XB² + 56.25) = 6.25 XB

6.25 XB² + 56.25 = 39.0625 XB²

32.8125 XB² -56.25 = 0

XB = 1.3093073414159541 or 1.309 (rounded)

So line AX = 4 - 1.309 miles = 2.691 miles and
Line XC = 3.273 miles (Square Root of (1.309² + 3²)
Cost of Pipe along Line AX = 2.691 million dollars
Cost of Pipe along Line XC = (3.273 * 2.5) = 8.1825 million dollars
Total Cost = 10.8735 million dollars.

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