# Flow Time in an Hourglass

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This Demonstration shows how the flow of sand in an hourglass (sand timer) changes as you vary the radius of the neck between the two cones.

Contributed by: Alexandra Baumgart and Hazuki Okuda (June 2015)
Open content licensed under CC BY-NC-SA

## Details

The equation of the volume of the frustum of the sand in the top cone with varying height is . Solving for , the height of the sand in the top cone in terms of volume is

.

The volume varies with the square of the neck width.

Snapshot 1: the width of the neck is very small; the top of the hourglass takes 33.5103 seconds to empty

Snapshot 2: the width of the neck is very large; the top of the hourglass takes 0.930842 seconds to empty

Snapshot 3: the width of the neck is intermediate; the top of the hourglass takes 2.58567 seconds to empty

References

[1] Paul's Online Math Notes. "Calculus I - Notes." (May 15, 2015) http://tutorial.math.lamar.edu/Classes/CalcI/RelatedRates.aspx.

[2] Math Central. "Sand in an Hourglass." (May 14, 2015) http://mathcentral.uregina.ca/QQ/database/QQ.09.09/h/luke1.html.

## Permanent Citation

Alexandra Baumgart and Hazuki Okuda

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