cord = 128 cu.ft. = 14.22 cu.yd.
So if the dump truck has a 14 cu.yd. box (not uncommon), then something approaching three cords is not unreasonable..
I like your method better than mine, but since I already started typing, here's how I do it:
D = Average diameter of log in inches
L = average length of logs in feet
N = number of logs
A = pi/144*(D/2)^2 = average cross-section in sq.ft.
V = A*L*N = total volume of wood in cu.ft.
cords = V / 85
Most fuel calculators estimate a cord at 85 cubic feet of solid wood (128 cu.ft. stacked), which is the basis of that last equation. Knowing pi/144 is roughly 1/46, one can usually do this pretty quickly in their head. Here's an example, not using your log truck, as I can't really guess these numbers from the photo, but just some hypothetical load:
D = 12 inches
L = 20 ft
N = 30 logs
A = 36/46 = approx 0.8 sq.ft. (or close enough in my head, when I'm standing outside in the cold)
V = 0.8*600 = 480 cuft.
C = 480 / 85 = about 5.5 (since 5*85 would be 425 and 6*85 would be 510)
You can do the math more accurately with a calculator, but since the starting numbers are just averages/guesses anyway, I don't see much point in busting out the slide rule.
Again, TreePointer's way is better, but this works in the field when it's not on a truck... which is usually where I'm doing it.