Chapter 4.5, Problem 49E

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

1. a. A plausible function to describe how gas mileage (in mi/gal) varies with speed v is g(v) = v(85 − v)/60. Evaluate g(0), g(40), and g(60) and explain why these values are reasonable.
2. b. At what speed does the gas mileage function have its maximum?
3. c. Explain why the formula C(v) = Lp/g(v) + Lw/v gives the cost of the trip in dollars, where L is the length of the trip and v is the constant speed. Show that the dimensions are consistent.
4. d. Let L = 400 mi, p = $4/gal, and w = $20/hr. At what (constant) speed should the vehicle be driven to minimize the cost of the trip?
5. e. Should the optimal speed be increased or decreased (compared with part (d)) if L is increased from 400 mi to 500 mi? Explain.
6. f. Should the optimal speed be increased or decreased (compared with part (d)) if p is increased from $4/gal to $4.20/gal? Explain.
7. g. Should the optimal speed be increased or decreased (compared with part (d)) if w is decreased from $20/hr to $15/hr? Explain.