Challenge Level

**Why use this problem?**

This problem is based on real-life biology, and also includes the concept of half-life. Students will need to decide how to use the information that they are given, and will need to do some estimating.

The idea of bounds is very useful here. Students can try to find a number for $X$ which they know is an underestimate. They can also try and find an overestimate for the volume of Rudolph's nose.

You can download a word and pdf version of this problem.

**Key Questions**

- What do we know? What do we not know, which we need to know?
- How might we estimate what we don't know?
- How many bacteria do we have after 30 minutes? How many after 1 hour? How many at the end of the 24 hours?
- If we have $n$ bacteria, how many molecules of $X$ will have been emitted during a 10 minute period? Can you find an underestimate for the number of molecules of $X$ left at the end of the 10 minutes, assuming that the half life is 10 minutes.
- If we think of the molecules of $X$ emitted near the start of the 24 hours, most of them will have decayed by the end of the day, which means we can ignore any early molecules. At what point should we stop ignoring them?

This website gives the sizes of various sports ball which might be useful.

**Possible Extension**

Can you find an upper limit for the size of Rudolph's nose?