A needle in a haystack
by Trevor Moore
I expect that you have seen this picture (courtesy WA’s Department of Fire & Emergency Services) if you are a reader of the Australian news. For those who are not, I will explain. Somehow or another a radioactive item with the dimensions shown above fell off a lorry as it was being conveyed - presumably with other items - from a Rio Tinto mine site, north of Newman in the Pilbara region, to Perth, where it was being sent for repair. There are many questions that should rightly be raised about this but this is not what I have come to talk about. Over the last few days people drove slowly along the 1,400 kilometre route that should have been taken by this pesky cylinder with a Geiger counter to see if it can be located.
Today these people were successful in their quest and I heard a newsreader say that this was like “looking for a needle in a haystack” and I wondered whether it was indeed like looking for a needle in a haystack, so I decided to do some calculations.
Somewhere in this haystack is a needle. Let us find it.
The first thing one needs to determine is how big a haystack is and the second is how big a needle might be. I have had to make some assumptions. For the haystack I found a picture of a traditional English haystack and I estimated its volume. I have reproduced the picture here. You will see that the haystack is traditionally shaped: it is a cylinder with a conical top. I have estimated its dimensions with reference to the fencing that surrounds it that I have supposed is about 1.2 metres in height. I have assumed that diameter of the haystack to be 4 metres (a diameter of 2 metres), the height of the cylindrical piece to be 3 metres and the height of the conical section to be 2.5 metres. The formula for the volume of a cylinder is V = (𝛑r2h1) and for a cone is V = (πr2h2)/3 where in each case r = 2 metres, and h1 = 3 metres and h2 = 2.5 metres. This gives me a total volume of 48.17 m3.
Spending an hour at one metre’s distance from the missing capsule is similar to receiving 10 X-rays. This x-ray is of my bionic hip.
Now for the needle. This is a tricky as we need to make some particular assumptions about whether the needle that was lost was being used for sewing a hem on a cotton frock, for darning the farmer’s socks or for creating a tapestry to make a wall hanging. We will go with the first of these and assume that we are using a size 24 needle. Now, Musers, I can tell you that a size 24 needle is 0.24 mm in diameter, that is its radius is 0.12 mm. I discovered this after extensive research and let me tell you it took a while to discover – even though I had guessed as much. Perhaps this is just meat and drink to a seamstress of the skill and application and experience of my sister. I do not know. I cannot say. Anyway, I then assumed that the needle that we have lost was 5 cm long and that, for estimation purposes, it is a cylinder. This gives me a volume of 2.26 mm3. A simple division tells me that I can fit 21,296,296 of these needles into my haystack.
Geiger counters have been pressed into service in the search for the little capsule. Well, of course they have. Now, we need to turn to the radioactive capsule. Here we can use area rather than volume without, as a mathematician would say, loss if generality (that’s not quite true but it will work for our purposes). So, we need to account for 1,400 km of road. I have supposed that the road is 10 metres wide. This does not take account of the fact that the radioactive capsule probably bounced around a bit – and if it fell off its transport at any speed it could well have bounced a fair way. Equally it could have been squashed into the bitumen by a following or passing vehicle. But work with me. We need to search an area of 14,000,000 m2. The radioactive capsule is 8 mm high and 6 mm across; it’s a cylinder but we are working with areas so we will take the cross section: 6 x 8 = 48 mm2. Another quick division tells us that – assuming all sorts of things about probability and whether the capsule landed on its side or on its end – there are 291,667,000,000 places that it could be on that road. This is bigger than 21,296,296 – as a physicist would say – by three orders of magnitude.
Ergo, looking for this radioactive capsule is 1,000 times more difficult than looking for a needle in a haystack.