Question

Andre, a physics undergraduate, enters a coffee shop that is holding a contest. Customers who guess the number of coffee beans inside of a closed, 20.0 fl oz aluminum commuter mug get a free pastry and an espresso drink. Motivated by the prospect of free food and coffee, Andre puts his finely‑tuned estimation skills to work.

Andre assumes that a typical coffee bean is about 1 cm long. Based on that assumption, what should Andre submit as his estimate for the number of coffee beans in the mug?

feedback says:

estimate the volume of the smallest rectangular solid that can contain a bean, then modify that result from the fraction that the bean actually occupies. Estimate the volume of air between the beans by estimating the fraction of the total mug volume actually occupied by the beans, use this volume in your calculation. You may have overestimated the value.

Answer 1

The length of each coffee bean is 1cm. Let us assume a cuboidal structure which can hold this coffee bean, then the structure must have a thickness of approximately half the length, ie 0.5cm since the thickness is approximately half its length.

Now another astute observation is that the breadth of this structure should be even less that the thickness, let it be 0.3cm.

Then the volume of one coffee bean will be,

The entire volume of the cup is,

It is only fair to assume that there must be some volume lost due to air between the coffee beans, lets be generous and say 20% of the volume is occupied by air and only 80% is occupied by the coffee beans. Then the volume occupied by the coffee beans is:

Then the number if coffee beans in the mug is given by: