The Algebra Symposium: Fuel Efficiency

The Problem

It is asserted that if you drive a car at constant speed; the number of miles per gallon will first increase and then decrease (or a reasonable range say 15 mph to 90 mph). Can someone explain the physics of this? Is there any sense that this function is quadratic?

After I did the note that follows, we realized that I was showing that gallons per mile , gpm (sic), "will first decrease and then increase." Actually we got the idea listening to the (e-mail) discussion of the nonlinear term - showing the value of cooperative work!

Minimization of Total Consumption for Any Convex Nonlinearity

If v = velocity, M = miles, E(v) = time rate of fuel consumption at speed v, then the total time T to travel M miles is M/v, and the total fuel consumed, G, to travel M miles is

G = M v * E(v).

Assume that E(v) = E₀ + E₁ v + ϕ(v), where ϕ(v) is the nonlinear part . It is reasonable to assume that E(v) is a convex (nee concave up) function of v. The following observations apply to any ϕ(v) which is convex.

As we all know from Math 165, E(v)/v, is minimized when

E′(v) = E(v) v ,

by the quotient rule for differentiation.

Geometrically, this is seen drawing the graph of E(v), and then observing that the slope of the line from 0 to (v,E(v)) has slope E(v)/v; the slope is minimized when this line is tangent to the graph.

The result in Math 165 microeconomics is that the Average Cost of producing the first q units, C(q)/q, is minimized when it is the same as the Marginal Cost , C^′(q), of producing the qth unit:

A(q) = MC(q).

Units Note

The original problem was about miles-per-gallon which is a constant times [M/G] = [v/E(v)]. Thus minimizing E(v)/v is the same problem as maximizing miles-per-gallon.

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On 24 Aug 2014, 16:01.