Travel costs A simple model for travel costs involves the cost...

Question

Travel costs A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume gasoline costs $p/gallon and the vehicle gets g miles per gallon. Also assume the driver earns $w/hour.

e. Should the optimal speed be increased or decreased (compared with part (d)) if L is increased from 400 mi to 500 mi? Explain.

e. Should the optimal speed be increased or decreased (compared with part (d)) if L is increased from 400 mi to 500 mi? Explain.

Accepted Answer

Welcome back, everyone. In this problem, a company is planning to send a shipment over a distance of 300 miles. The truck consumes diesel at a rate described by the function D of V equals V multiplied by 90 minus V divided by 70, where D is the diesel efficiency in MPG and V is the speed in MPH. The diesel costs $3 per gallon and the driver's wage is $25 per hour. Should the speed that minimizes the cost be increased or decreased if the distance is increased from 300 miles to 400 miles? Justify your answer. A says it should be increased because a longer distance means a larger wage for the driver. B says it should be decreased because at a longer distance, the fuel efficiency is higher at lower speeds. C says it should be increased because a longer distance means a greater fuel efficiency which improves at higher speeds. And the D says it should be neither increased nor decreased because the speed that minimizes the cost remains the same regardless of the distance. Now if we're going to figure out. The cost, OK? If the speed that minimizes the cost be increased or decreased for an increase in distance, essentially we're trying to figure out our total cost as a function of the speed, OK? And then we're trying to understand how the distance affects that total cost, given it's a function of the speed. Now, if we're gonna do that first we need to figure out or we need to think about the total cost for the shipment. What's involved here? Well, we know that the total cost, cost is going to be made up of two things. So let's call that total cost CFee. First, it's gonna be made up of the fuel cost, OK, because it's going to cost money to um buy our fuel. And then we also know that it's going to cost us to pay the driver, so we have the driver's wage cost. So let's try to express both of those costs in terms of our function D of V or in terms of the speed V, and then we can better understand what's involved for a total cost to help us minimize it with respect to our speed V. So first, let's think about our fuel cost. No I feel the cost in. is basically going to be equal to the um the cost per gallon for fuel, OK. Multiplied by the number of gallons used. OK. The number of gallons used. Now what do we know about both of these quantities? Well, our problem told us that fuel costs $3 per gallon. So the cost per gallon is $3 OK? Next, we know that for the number of gallons used it's going to be equal to the total distance divided by the speed. OK, divided by the, the diesel efficiency, sorry, divided by the diesel efficiency. So that's going to be equal to our length, let's call that length L, divided by D of V, which means it's going to be multiplied by 70 divided by V multiplied by 90 minus V. OK. Which, when we expand that, the fuel cost is gonna be equal to 210 L divided by V multiplied by 90 minus V. So this is our fuel cost, OK? Next, what about our wage cost? What do we know about that? Well, our wage cost. It's gonna be equal to the amount of money we have to pay the driver per hour. OK, in other words, the driver's hourly wage. Multiplied by the time that the driver spends on this on this journey. OK, now we know in our problem the hourly wage is $25 per hour, and to find the time, it's gonna be equal to the distance L divided by the speed at which the average speed at which the driver is moving, OK, or the truck is moving rather, and this time is going to be in hours. So that means then that the wage cost is gonna be equal to 25, OK, divided by Z. Know that we have our wage cost and fuel cost in terms of the, then we can we can solve them. Let me put that in red. We can solve for our total cost as a function of V, which is gonna be equal to. 210 l divided by v multiplied by 90 minus V plus 25 L divided by V. Notice here that we can factor L from both our terms. Now that's gonna leave us with 210 divided by V multiplied by 90 minus V plus 25 divided by V. Now when we write our total cast in this way, what do you notice? Well, here, as we can tell, L is just a multiplicative constant, OK? So that means then. That if we want to find a cost per mile. And we don't really need L. We can find the cost per mile as a function of our speed, V, which is gonna be equal to 25 divided by V plus 210. Divided by V multiplied by 90 minus V, essentially what we had here inside our brackets. And know the speed that will minimize the cost can be obtained by taking the derivative of this function and setting it to zero. So since the factor L multiplies the cost per mile, and that means the function that we are minimizing 25 divided by V plus 210 divided by V multiplied by 90 minus V is independent of the distance L. In other words, we don't need it to minimize our function. So that means whether the shipment is 300 miles or 400 miles long, the value of V that minimizes the cost per mile and hence the total cost remains the same. Therefore, if we look back on our answer choices, that means D is the correct answer. It should neither be increased nor decreased because the speed that minimizes the cost remains the same regardless of the distance. Thanks a lot for watching everyone. I hope this video helped.

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