Minimize the cost of travel for truck driving.

Question

A truck has a min. speed of $9mph$ in the highest gear. When traveling $x$ mph, the truck burns diesel fuel at a rate:

$0.0224868((1156/x) + x)$ gal/mile

The truck can not be driven over $51mph$ and the diesel fuel costs $1.44 gal, find the following.

(a) The steady speed that will minimize the cost of diesel fuel for a 470 mile trip.

The answer for (a) is 34 mile/hour, this is the min. of the function above found via the Extreme Value Theorem.

This is where I am having trouble:

(b) Find the steady speed that will minimize the cost of a 470 mile trip when the driver is being paid $17/hour.

• I have found out that total cost = (cost of the driver) + (cost of fuel for the trip)

I have yet to put any real work behind this because I am lost and I do not know where to start to figure this problem out. I could try a bunch of values, but that would not work and take so long. I thought about making the bounds [34, 51] and somehow finding the inflection point on that, but I do not believe that would work.

I'd appreciate some help with this problem. Thanks.

Answer 1

If you solved 1), solving 2) isn't much harder. Just express the cost of the driver as a function of the distance and the speed $x$. Then solve the problem using the same technique as 1).

Answer 2

This is an interesting problem so I'll give my thoughts on it. Suppose we have a truck that has a minimum speed of $v_{\min}$, a maximum speed of $v_\max$, and when moving at a rate $v$ consumes gas at a rate $g(v)$ gallons per mile, as in the question, although gallons per hour would be a more realistic measure. Say that the fuel costs $c$ dollars per gallon, the driver is paid $p$ dollars per hour, and we want to minimize the cost of traveling $d$ miles. Assuming we pick a constant velocity $v$ to travel the entire trip, here is how we compute the total cost $C(v)$:

The amount of gas consumed will be the distance of the journey in miles multiplied by the amount of fuel consumed per mile, that is $$\text{gas consumed}=d\cdot g(v) \text{ (gallons)}$$ Then, using the cost of gas per gallon: $$\text{cost of gas}=d\cdot g(v)\cdot c\text{ (dollars)}$$ Now we need to determine the amount the driver is paid for the trip, which is his hourly rate multiplied by the number of hours the trip takes. $$\text{driver pay}=p\cdot \frac{d}{v}\text{ (dollars)}$$ The total cost is the sum of these two. Hence, $$C(v)=d\left(c\cdot g(v)+\frac{p}{v}\right)$$ Minimizing this function in the interval $[v_\min , v_\max]$ with respect to $v$ will give the answer.