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 per gallon and the vehicle gets ?g? miles per gallon. Also, assume that the driver earns $w/hour. a. A plausible function to describe how gas mileage (in mi/gal) varies with speed is g?(???)= ???(85??)/60. Evaluate g(0), g(40), and ?g?(60) and explain why these values are reasonable. b. At what speed does the gas mileage function have its maximum? c. Explain why the cost of a trip of length ?L? miles is ?C?(?v?)= ?Lp/g?(???)+ ?Lw/??. d.Let ?L? =400 mi, ?p? =$4/gal, and to = $20/hr. At what (constant) speed should the vehicle be driven to minimize the cost of the trip? e. Should the optimal speed be increased or decreased [compared with part (d)] if ?L is increased from 400 mi to 500 mi? Explain. 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. g. Should the optimal speed be increased or decreased [compared with part (d)] if ?to is decreased from $20/hr to $15/hr? Explain.
Solution Step 1 Consider that a simple model for travel costs involves the cost of gasoline and the cost of driver. It is assumed that the gasoline costs per gallon and the vehicle gets miles per gallon. Also, it is assumed that the driver earns per gallon. Step 2: (a)A function to describe how gas mileage (inmi/gal) varies with speed is given as g(v) = v(85 v)/60 0(850) g(0) = 60 = 0 And, Also, Hence, g(0) = 0 g(40) = 30 g(60) = 25 These values are reasonable because the function shows the variation of the mileage with the speed, and corresponding to these mileage values, the speed values are optimum that is no speed, speed of40 mi/hr, and speed of60 mi/hr Step 3 (b)To find the speed at which the gas mileage function has its maximum, differentiate the mileage function with respect to the speed, and equate it to zero. Equate this to zero Hence, the speed of at which the mileage has the maximum value is42.5 mi/hr Step 4 (c)It is given that the cost of a length of trip miles is given by the formula C(v) = L p + L w g(v) v Now, the cost of a trip of length, say 1 miis given by sum of the total gasoline consumption and the fare of the driver Now, for length of the total gasoline consumption is given by the cost of gasoline divided by the mileage, that is p g(v) And the total fare of the driver is given by amount driver earns per hour divided by the speed, that is w v Thus, the total cost comes out to be total gasoline consumption+total fare of the driver= p + w g(v) v Now, this was the cost when the length of the trip was1mi Therefore, for a trip of length miles, the cost of the trip is given by p p ( g(v)+ v = L g(v)+ L v Hence, the cost of a length of trip miles is given by the formula: C(v) = L p + L w g(v) v (d)Consider the following Substitute the values in the function of cost p w C(v) = L g(v) + L v 96000 8000 = v(85v)+ v To find the speed at which the vehicle be driven so as to minimize the cost of the trip, differentiate the cost function with respect to , and equate to zero. d d 96000 8000 dv(C(v)) = dv( v(85v) + v ) = 96000(852v) 8020 v (85v) v 96000(852v)8000(85v) = 2 2 v (85v) Now, equate this to zero Solve the above quadratic equation to get the values of , which in this case are: However, the speed at which the cost shall be low will be at the lower speed. Hence, the value of the speed at which the cost shall be low is 62.885 mi/hr