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. 

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.

b. At what speed does the gas mileage function have its maximum?

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.

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?

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 w is decreased from $20/hr to $15/hr? Explain.