# Big and small numbers in the living world

Biology makes good use of numbers both small and large. Try these questions involving big and small

- Estimate how many breaths you will take in a lifetime.

- Estimate the number of people who live within 1 km of your school.

- I look through a microscope and see a cell with a roughly circular cross section of 7 microns. Estimate the volume of the cell.

- The arctic tern migrates from the antarctic to the arctic. Estimate how far an arctic tern flies in its lifetime.

- Estimate the weight of dog food eaten in England each year.

- An amoeba doubles in size every 24 minutes. How long will a sample of size about 1mm by 1mm take to cover a petri dish? Do we need to worry too much about the initial size of the sample in this calculation?

- Estimate how many cells there are in your little finger

- Estimate the total weight of the pets of everyone in your class

- How much do you think your brain weighs? How much do you think this weight varies over the course of your life?

- If current trends continue what will the population of the UK be in ten years' time? Do you think it is reasonable to assume that they will continue? Why?

Extension: Try to improve your estimates by giving a range of numbers between which you know the true answer lies.

Although there is no 'right' answer to an estimation, there are good or bad estimations and sensible or over detailed calculations.

Think how you might make your estimation a good one, and think how it makes sense to ignore certain complexities in the calculation.

**1. How many breaths will you take in a lifetime?**

You might notice that you breathe once every 5 seconds. This means 12 breaths per minute, which is 720 breaths per hour, or 17280 breaths in a whole day.

Now, the average person lives 80 years, i.e. $ 80 \cdot 365 = 29200 $ days (or $29220$ days if you want to count the 20 leap years in between!)

So, the number of breaths you will **on average** take in your lifetime will be $29200 \cdot 17280 = 504576000 $!

Of course, you need to take into account the fact that babies usually breathe faster, and also that during or after physical exercise you inhale and exhale a faster as well. This could certainly increase our estimation by a considerable amount.

**2. How many people live within 1km of your school?**

The area concerned is a circle centered at your school, with radius 1km - it is quite straightforward to see that this area is $\pi \mathrm{km}^2 \approx 3.14 \mathrm{km}^2 $.

A simple assumption to make here is to consider the average population density of the UK (242 people per km$^2$) and multiply this by $\pi$. However, this is a relatively poor estimation. What you want to consider is the *urban* population density of the UK, which is obviously much higher (in fact 4348 people per km$^2$).

Then, an estimation of the number of people living within 1km of your school would be $$3.14 \cdot 4348 = 13653 $$

**3. Volume of a cell.**

If the cross-section of the cell is roughly circular, and we assume that the shape of the cell is spherical, then its radius is 3.5 microns - so its volume will be

$$ V \approx \pi \cdot 3.5^3 \mu \textrm{m}^3 \approx 10.99 \mu \textrm{m}^3 $$

**4. How far does the Arctic Tern fly in its lifetime?**

An arctic tern lives for about 30 years (being one of the longest living birds). Every year, it migrates from Arctica to Antartica in order to see two summers, and hence more daylight. The distance between Arctica and Antartica is 19000 km, so two such journeys a year mean that the Tern flies 38000 km per year, and hence 1,140,000 km in its entire life!

In fact, though, the Arctic Tern does not fly in straight lines, but rather along the coasts, so recent research shows it flies up to 70000km each year! See http://en.wikipedia.org/wiki/Arctic_Tern for more.

**5. How much dog food is eaten in England each year?**

First, we will calculate how many household dogs there are. It is estimated that one out of four households in the UK owns a pet (think about how many of your close friends own one in order to convince yourself). Also, given that there are approximately 17 milion families in England, we see that there are on average 4.25 milion domestic dogs.

Now, let's assume that an average dog needs a 500g can of food each day (of course there are several alternatives, such as dry food, and also it is important to remember that the amount of food greatly varies with the dog's age, type and weight).

Then, under that assumption, English dogs consume $$ 4.25 \cdot 0.5 \cdot 365 = 775,625,000 \textrm{ kgs} $$ of dog food each year!

**6. How long will it take for an amoeba to cover a petri dish?**

A standard petri dish would have an area of about 100cm$^2$.

The amoeba, covers initially an area of 1mm$^2$, and it doubles every 24 minutes. This means that after $k$ periods of 24 minutes, the area of the amoeba will be

$$ A_k = 2^k \textrm{ mm}^2 $$

Now, 100cm$^2$ = 10$^4$mm$^2$.

We wish to calculate $k$ for which $2^k>10^4$. By considering various powers of 2, we see that $2^{13} = 8192$ and $2^{14} = 16384 $. So, we require 14 periods of 24 minutes, or (less than) 5 hours and 36 minutes, for the amoeba to cover the entire petri dish.

**7. How many cells do you have in your little finger?**

We will assume that the human cells occupy an area of about 10$\mu textrm{m}^2$. Now, your little finger should be about 5 cm long and 1 cm wide, so its area is 5cm$^2$. So, we divide the two to obtain that the number of cells in your little finger is about

$$ N = \frac{5 \cdot 10^8 \mu \textrm{m}^2}{10 \mu \textrm{m}^2} = 5 \cdot 10^7 $$

**8. Estimate the weight of all your classmates' pets.**

We expect that 1 classmate out of 4 owns a dog and 1 classmate out of 4 owns a cat. We also estimate that an average dog weighs 25 kg (although the range of possible dog weights is huge, so this estimation is not very well-founded) and that an average cat weighs 5 kg (the weights of cats are a lot more concentrated around this value).

Therefore, if your class has 20 pupils, the total weight of all your classmate's pets should be

$$ W = 5 \cdot 5 + 5 \cdot 25 = 150 \textrm{kg} $$

**9. How much does your brain weigh?**

In general, the brain of a newborn baby weighs about 300 grams (so is about 5% of the baby's entire weight). As the human is growing older, though, the brain's weight becomes a smaller percentage - in particular, an adult's brain weighs approximately 1,5 kgs, which is around 2% of the whole body's weight.

**10. What will be the UK's population in 10 years' time?**

The UK's population in 2001 was 58.8 million. The census of 2011, which was only recently carried out has not produced official results yet, but it is expected that the UK's population will appear to be around 62.4 million.

Now, if we are to assume that the population grows at a constant rate, we see that it should increase by 3.6 million every 10 years, so we would expect that in 2021 the UK's population should be approximately 66 million people.

However, over the course of 10 years, many factors which affect a country's demographics change, such as the economic situation in the UK and abroad, the likelihood of finding employment compared to other countries, environmental issues etc. So, the rate of change of the population is almost never the same over two decades. In fact, the tendency is that the rate of change increases, (this is what has been happening from 1981 onwards) and so a better estimate would be that the UK's population would significantly exceed 66 million by 2021.

### Why do this problem ?

Practice with the use of numbers is a crucial biological skill. These interesting questions will allow you to practice these skills whilst developing awareness of orders of magnitude in scientific contexts.### Possible approach

There are several parts to this question. The individual pieces could be used as starters or filler activities for students who finish classwork early. Enthusiastic students might work through them in their own time. Since there is no absolutely 'correct' answer to many of these questions, they might productively be used for discussion: students create their own answers and then explain them to the rest of the class. Does the class agree? Disagree? Is there an obvious best 'collective' answer?### Key questions

- What assumptions will you need to make in this question?

- How accurate do you think you answer is?

- What order of magnitude checks could you make to test that your answer is sensible?