While Fermi problems are typically to do with some real world context, some can be purely mathematical.

How many tennis balls would fill your classroom?

Estimate the number of tennis balls that would fill a classroom
.

This is a basic problem: it can be refined and extended by changing the dimensions of the room and using different kinds of balls.

**Possible approach**

Assumptions needed:

- Shape of classroom – cuboid?
- All balls are the same size

How the balls are arranged:

- In parallel identical layers?
- Another pattern?

Data needed?

- Typical room dimensions
- Diameter of tennis ball

**Possible solution**

Obviously this involves more than a simple ratio of volumes!

Diameter of ball (d = 6.7 cm)

Suppose the room has linear dimensions (x = 8m, y = 6m , z =4m)

Nominate a prior (guess)estimate before proceeding.

How does an estimate of 620 000 look? How good was the original estimate?

Using a simple ratio of the volume of the room to the volume of the ball gives a result that is about double the estimate. Is this fair comment? Why is this so? Will it always be the case?

Change the dimensions of the room. What if we use golf balls? Soccer balls?

**Some pedagogical notes**

- Students should identify necessary assumptions, and provide the dimensions of room and ball.
- They should also contribute reasons why a simple ratio of volumes of room to ball is not appropriate.
- Since 800/6.7 = 119.40 etc the number of balls actually fitting along a linear dimension would be given by rounding down. But does this matter in calculating an estimate? (It makes about a 2% difference to the answer obtained).
- Volume of sphere = 4/3∏r
^{3} gives volume_{room} / volume_{ball }= 6xyz/∏d^{3 }
- This is (6/∏) ≈ 2 times the estimate and will always apply as it doesn’t involve x, y, z or d.
- Changing dimensions of the room and/or the ball will give insights about the impact of changes in linear dimensions upon corresponding volumes.

**References**

Adam, John, A. (2003). *Mathematics in Nature: Modeling Patterns in the Natural World*. Princeton: Princeton University Press.
Arleback, Jonas. (2009). On the use of Realistic Fermi problems for introducing mathematical modelling in school. *The Montana Mathematics Enthusiast*, *6 (3), *331 -364*. *