The range of an aircraft is how much distance it can cover with the available fuel. The last post I did on range covers the air range - how far the airplane can go through the air. But the air is usually moving, so the air range isn't the same as the ground range. If there is a fifty knot [nautical mile per hour] tailwind at the flight altitude, then over the course of an hour, the airplane will travel fifty miles further over the ground than through the air. And if it's a fifty knot headwind, the ground range will be fifty miles less than the air range, for every hour of flight. And of course it's the ground range that determines whether you get to Anchorage before the fuel gauges get to E.
As soon as there is a headwind, ground range will always be less than air range, but one can somewhat offset the negative effect of a headwind by speeding up. The handwaving argument says that even though increasing the fuel flow will increase fuel consumption per air mile, increasing the ground speed will minimize the fuel flow per ground mile.
The easiest way to prove that an increase in speed could increase range, is to consider the extreme case. Imagine that the airplane is travelling at its best range speed, at a true air speed of 200 knots, but it's in a 200 knot headwind. The effective ground range is zero. If the fuel will last four hours at this fuel flow, then after six hours, the airplane will have gained no ground. Increase the fuel flow such that the fuel on board will only last five hours, and the airspeed will increase. It doesn't matter how small the increase in airspeed is from that increase in fuel flow, it translates to a positive groundspeed over those five hours: maybe it's going forward at 20 knots, for a total of 100 miles range. Increasing the fuel flow further, so the fuel would only last four hours would increase the speed further, but if the groundspeed at the new fuel flow was 24 knots, then the range over the four hours would only be 96 miles.
The trick is working out how much to increase the fuel flow over best air range to offset the wind. Here's the theoretical method using the fuel flow versus true airspeed graph discussed earlier. Look at the x-axis, the airspeed axis. Change the scale so that instead of true airspeed it reads in ground speed. So if there's a fifty knot headwind, the zero knots point of the original graph becomes minus fifty, the fifty knot point of the original graph becomes zero, and the hundred knot point of the original graph becomes fifty. Basically you're subtracting the headwind from every notch of the scale. Now, and this is slap-myself-in-the-forehead obvious but I didn't think of it: find the best ground range by drawing a tangent from the groundspeed zero (the true airspeed fifty knots) to the thrust required (fuel flow) curve.
In the airplane, I know how to use that theory to work out a practical best range speed by making a little table of fuel flow versus airspeed, and I know the approximate zero-wind best range speed (it varies with loading, altitude and temperature) of the aircraft I fly, and then using a rule of thumb to compensate for the wind. I can now see that I should be using the GPS groundspeed to obtain the best range speed. In a large jet, the weight change with fuel consumption is significant enough that for optimum range, the best range speed needs to change as the weight decreases. I imagine Dave has an onboard computer that compares his INS groundspeed with the metered fuel flow and can tell him the best range, as well as the most economic speed to fly at every moment. I don't have one of those, but I'm looking forward to learning all about how to use one.