Attn: Mathematicians! A truckload of dimes or quarters?

Which would yield more cash? A friend of mine says quarters. I say dimes.

Curiously, according to this site dimes and quarters, measured to a thousandths of a gram, are worth the exact same per gram. (Dismissed as coincidence?)

I say then that dimes win, due to the lower volume. He says “but you need 2.5 dimes per quarter to match.”

Good point, says I.

Says he, we buy a graduated cylander, and measure the volume.

I don’t think it will answer the question though. Sheer volume will not take size and shape into account, will it?

I think this requires a bit of calculus. I have a vague idea of what calculus is for, but know nothing of its practice.

I know there are some crack mathematician/engineer types here. Can any of you help me win a bet?

sorry, you lose.
but just barely, they are very close.
As you said, the mass per dollar is exactly the same for dimes and quarters. So if the truckload is weight limited they tie.
If the truckload is volume limited, the quarters win. since the two objects are in the same shape, you dont have to worry about optimal packing schemes or such. just compare the thickness * diameter^2, which will be proportional to volume per dollar.
dime is 1.35mm thick with 17.91mm diameter.
Quarter is 1.75mm thick with 24.26mm diameter.
so dime takes up 5.11% more volume for the dollar.

Isn’t that the same as using the graduated cylander?

Optimal packing scheme is exactly why I think some calculus is required. I’m not saying you’re wrong, you just haven’t convinced me.

How about a truck packed full of quarters, with dimes strategically placed between the cracks of the stacks of quarters?

Myself, I’d pick a truck full of sackies, myself :smiley:

Maybe I should learn to use preview. 2 Myselfs in 1 sentence just doesn’t sound too good. Oh well.

They have the same shape (short cylinders), so, in the long run, they would have the same optimal packing scheme. I’m pretty sure that it is layers of coins laid side to side in hexagonal packing. The dimensions of that packing scheme would depend on the primary dimensions–and it would seem that the dime would require more space for equivalent value.

When dimes and quarters were silver, it would make sense that a quarter would have exactly 2.5 times as much silver. Clearly, the same sort of ratio has been maintained for the cupro-nickel clad coins. The dime must be “thicker” than necessary, because of the effects of the image relief, which are more pronounced for the smaller coin.

Assuming that the coins are arranged neatly in layers then the relative thickness will not matter. You will have more layers of Dimes but so what? Think of the columns of coins as cylinders. Optiomal packing density (hexagonal array) will be the same for both when you ignore the boundaries. Unless you can cut coins in half you get wasted space where the layer meets the side of the container. This effect is greater for larger coins so Dimes win.

BTW you do not need calculus, just geometry.

I disagree. I think you’d be correct if the coins had the same value per volume, but, according to Luckie, they don’t: quarters have more value per volume by roughly 5%. Since a reasonably-sized truck has interior dimensions on the order of 2.5 meters (100X a quarter diameter), I doubt that the dime’s “edge effect” advantage overcomes the quarters’s value/volume advantage, for an optimally-packed configuration.

OK, question: Why are we talking about optimal packing density? What seems more interesting, and more in the spirit of the OP, is: Which is worth more, a truckload of loose dimes or a truckload of loose quarters? (Gatsby? Does your bet specify loose or optimally packed?)

I’m thinkin’ that the packing density of loose coins might be a function of the diameter-to-thickness ratio, in which case the lower D/t ratio of the dime might offset the higher value/volume of the quarter. Or do loose coins tend to sort themselves into a near-optimum packing configuration automatically? Anyone?

That will teach me to post before properly reading all those before me. My only defence is that I am at work and really should have my mind on other things.

Thinking back to my bucket of loose change at home I think that the coins tend to lie flat more or less in layers. They are unlikely to be in an optimal configuration but I would expect that the amount of wasted space is about even for both coins.

Were the loading onto the truck very fast then the configuration would be less planar than my change bucket, where I just toss in a few coins at a time. The probability that a significant number of coins were not flat would, I think, increase with the loading rate.