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A Sand Story-Fact or Fiction by Ed Gallo

The other day, I was talking to an old friend about how his business was doing. Joe told me that things were fine with his hauling business.  He and his 8-yard dump truck were busy all the time. Then, Joe told me that he delivered a load of very fine sand to a customer who wantedthe sand dumped in the middle of his wide, flat driveway. When Joe got out of the truck to see if all of the sand was dumped, he noticed (like he noticed a thousand times before) that the sand was piled up in a somewhat conical-shaped pile. Joe told me that he wondered about the measure of the acute angle betweenthe slant height of the sand pile and the horizontal driveway. Interesting!

But, Joe continued by telling me that on the same day he made a second
delivery of a course sand to another customer. He dumped the sand, then
took a look at the pile before he drove away. Again he wondered about that acute angle. He did notice
that the acute angle was not the same as the pile of very fine sand that he had dumped earlier. Very

Oh, but Joe then related how he made a third delivery of number 8 stone. (Number 8 stones
are pea-sized stones from ¼ to ½ inch in size.) Again, he dumped the stone, observed the stone pile,
observed the acute angle between the slant height of the stone pile and the horizontal driveway,
and drove away.

Joe finished his story by commenting that the acute angle appeared to be different for piles of different materials. And, finally, Joe asked me if there was a way to mathematically calculate the acute angle for a pile of very fine sand.

Now, this becomes even more interesting because I know that there is a mathematical way to find that acute angle! Oh, but what is it? I saw the development of the solution to this acute angle problem during a seminar by a Distinguished Visiting Professor at another college a long time ago. And, of course, the details of the mathematical work escapes me at this moment. So, I must close this Sand Story by offering a challenging to everyone. The challenge: Solve this acute angle problem for very fine sand and offer your solution in the next issue of the OhioMATYC Newsletter. Maybe your calculus students can help you with this!