Challenge Level

Imagine plotting a graph of $y=2^x$, with $1$cm to one unit on each axis.

How far along the $x$-axis could you go before the graph reached the top of a sheet of paper?

If you extended the graph so the positive $x$-axis filled the whole width of a sheet of paper, how tall would the paper have to be?

How far along the $x$-axis would you have to go so that the graph was tall enough to reach

- to the top of The Shard in London?
- to the moon?
- to the Andromeda galaxy?

Try to estimate the answers before calculating them and mark them at the appropriate points along a sketch of the $x$-axis.

Work out where they should be and then add some other results such as the distances to the sun and other stars. What do you notice?

We have provided some data below for you to work from, or you could research suitable data for yourself.

* A4 paper measures $298$mm by $210$mm.

* A3 is double the area of A4 --- $298$mm by $420$mm.

* To go from A4 to A3 and from A3 to A2 you double the shorter dimension each time. The sequence continues up to A0.

* American Letter size paper is $8.5$in. by $11$in..

* The Shard in London is $309.6$m high. Its viewing gallery is approximately $244$m above ground level.

* One of the tallest buildings on Earth is the Burj Khalifa in Dubai, at $829.8$m.

* The highest mountain is Mount Everest, at $8\,848$m.

* The mean distance from the Earth to the moon is $378\,000$km.

* The mean distance from the Earth to the sun is $149\,600\,000$km.

* After the Sun, the nearest bright star is Alpha Centauri, at a distance of about $4.4$light years from Earth.

One light year is the distance that light can travel in a year. Light travels at a speed of about $300\,000$km/s.

* Andromeda, the closest major galaxy to our own, is approximately $2.5$Mly from Earth.

$1$Mly (megalight year) is $10^6$light years.

* The edge of the known Universe is approximately $13.8$ billion light years away.