According to the story, which of the following are components of the total cost for the Chicago to New Orleans run?
least cost
[From Calculus&Mathematica]
You are the chief dispatcher for the C&M Trucking Company, which sends Mack trucks on the straight shot between Chicago and New Orleans on Interstate 57. You know that:
- The run between the two cities is \(750\) miles.
- Running at a steady \(50\) miles per hour, the Mack gets \(4\) miles per gallon.
- For each mile per hour increase in speed, the big Mack loses \(\frac {1}{10}\) of a mile per gallon in its mileage.
- The driver team gets \(27\) dollars per hour.
- Keeping the truck on the road costs an extra \(12\) dollars per hour over and above the cost of the fuel.
- Diesel fuel for the Mack costs \(\$3.09\) per gallon.
Approximately, what steady speed should you tell your drivers to hold in order to
make the run at least cost?
If the truck travels at a steady speed of \(v\), then how long with it take to complete the
trip?
time = \(\answer {\frac {750}{v}}\) hours.
If the truck travels at a steady speed of \(v\), then what will be the cost of the driving
team
Driving Team Cost = \(\answer {\frac {20250}{v}}\) dollars.
If the truck travels at a steady speed of \(v\), then what will be the cost of maintaining
the truck?
Truck Maintenance Cost = \(\answer {\frac {9000}{v}}\) dollars.
The final cost to calculate is for fuel.
Fuel cost \(\$3.09\) per gallon, therefore we need to know how many gallons the trip will take. To calculate gallons, we need the gas mileage for the truck.
If the truck travels at a steady speed of \(v\), then it is \(\answer {v-50}\) miles per hours over \(50\). For each mile per hour over \(50\), the truck loses \(\frac {1}{10}\) of a mile per gallon from \(4\) miles per gallon..
Therefore, it will lose \(\answer {(v-50)/10}\) miles per gallon when it is traveling at \(v\) miles per hour.
That gives a mileage of \(\answer {4-(v-50)/10} \frac {miles}{gallon}\).
The truck is traveling \(750\) miles and gets \(4-\frac {(v-50)}{10} \frac {miles}{gallon}\). From this we can calculate the total number of gallons needed, which will then give us the cost of fuel.
If the truck travels at a steady speed of \(v\), then what will be the cost of fuel?
Fuel Cost = \( 3.09 \cdot \answer {\frac {750}{4-(v-50)/10}}\) dollars.
That gives a total cost based on speed
\[ Cost(v) = \frac {27 \cdot 750}{v} + \frac {12 \cdot 750}{v} + \frac {3.09 \cdot 750}{4-(v-50)/10} \]
From the graph, we can see that there is a minimum cost of \(\$1161\) when the truck is traveling at a speed of approximately \(47.62\) miles per hour.
However, we would like an exact speed.
with Calculus
Calculus tells us that the derivative of \(Cost(v)\) is
\[ Cost'(v) = \frac {23175}{(90-v)^2} - \frac {29250}{v^2} \]
We can obtain the critical number by setting \(Cost'(v)\) equal to \(0\) and solving.
By obtaining common denominators and combining into a single fraction, we get a
numerator of
\[ \answer {23175} \, v^2 - \answer {29250} \, (90-v)^2 \]
We can multiply this out and gather like terms to get a quadratic equation
\[ \answer {6075} \, v^2 - \answer {5265000} \, v + 236925000 = 0 \]
There are two solutions, but only one is near \(47.62\).
\[ v = \frac {10}{3} \cdot \left ( \answer {130} - \sqrt {13390} \right ) \approx 47.62 \]
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