How many golf balls will be heard into the school bus?

For reference, the National Transport Vehicle Standards for Schools in the USA for 1995 indicate the maximum size of a school bus – 40 feet long and 8.5 feet wide. The standard diameter of a golf ball is 1.69 inches with a tolerance of 0.005 inches.


Clearly, this task is a kind of Fermi challenges when you need to provide an approximate estimation. Let’s take a look at one of the solution options.


A school bus, like any other vehicle, should correspond to the roadway parameters. It means it should not be much wider than passenger cars. You have definitely seen in the movies that there are seats for four children, as well as a passage in the middle. Also, there is a position where a teacher can stand. Let’s start from the point that the width of the bus is about 2.5 meters, the height is about 2 meters. You should understand that exact numbers are not so important as the order. How many rows of seats are there in the bus? Let’s assume that there are 12 rows. Each row needs about a meter or a little less, let’s take the length of 11 meters. Thus, the total volume will be approximately 55 mi.


Actually, the goalball diameter is approximately 3 cm. Let’s say that it is ~3.3 cm so that 30 such balls placed in a row will create 100 cm. The cubic design of 30x30x30 balls, that is, 27,000 balls, will fit in a cubic meter. Now, we multiply this by 55, it will result in about 1.5 million.


Note that many Fermi challenges are related to spherical sports items that fill buses, pools, planes, or arenas. You can gain additional points if you mention Kepler’s theory. In the late 1500s, Sir Walter Reilly asked the English mathematician Thomas Heriot to create a more efficient way of packing cannonballs on British military fleet ships. Harriot said his friend astronomer Johann Kepler about this challenge. Kepler proposed that the densest method of packing spheres is already used when laying cannonballs and fruit. The first layer is simply placed next to each other in the form of a hexagonal shape, the second in recesses at the junctions of the balls of the lower layer, etc. In large containers with this type of installation, the maximum density will be about 74%. Kepler believed that this is the densest packaging option, but could not prove it.


Kepler’s theory settled a largely unsolved issue for several eras. In 1900, David Gilbert created a popular list of 23 unsolved mathematical issues. Some people said they were able to explain this theory, but all their decisions were found unsuccessful and were supposed to be wrong. This lasted until 1998 when Thomas Heles proposed complex evidence using a machine that proved Kepler’s theory. Most experts are sure that its result will appear correct at the end, although its verification is not complete.


Earlier, we found that each golf ball actually lies in a transparent and thin plastic cube so that the cube edges are equal to the ball’s diameter. This indicates that the balls take up about 52% of the space (Pi/6, to be more precise). In case you grab the balls out of an imaginary cube, you can place a lot more balls in a provided volume, this is an empirically validated fact. Scientists have conducted experiments, filling large flasks with steel balls and calculating the filling density. The result was between 55% and 64% in space utilization. This is a denser option than we used, although it does not reach Kepler’s maximum, equal to about 74%. Additionally, the scatter of readings is pretty huge.


So, what should you do? We actually cannot place the balls ideally in real life, so it appears to be too absurd even to answer the absurd question. A much more practical goal is the density achieved by periodically shaking or stirring the container. You can achieve it if you distribute the balls with a stick more evenly. This will increase the density by about 20% than with the variant with a cubic structure. Thus, it is possible to increase the initial estimate to 1.8 million balls.


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