The Hornby Island soup can, and an elevator problem

 My first thought upon seeing this problem is that it is about proportional reasoning. It's stated in an interesting way: no specific values are given, and instead it's asking about a general solution process.

We need to determine the diameter and length of the tank. We can't really figure out the length right away, but from the picture it looks like the diameter is about three times the height of the bike.

Now, since we know the diameter of an actual soup can, we can calculate (height of bike)/(diameter of soup can) to figure out how many times taller the bike is than the soup can is wide. Since the width (diameter) of the tank is three times the height of the bike, the tank is 3*(height of bike)/(diameter of soup can) times bigger than an the actual soup can. This is our scale factor, so the length of the tank is (length of soup can)*3*(height of bike)/diameter of soup can).

Now, my first thought was to use these dimensions to calculate the volume, but we can also just use the scale factor directly. Since we're dealing with volume (scaling the can in three dimensions), the volume of the tank would be the volume of the can, multiplied by the cube of our scale factor.

The next question, whether it holds enough water to put out an average house fire, changes things up for two reasons. First, up until now, I haven't attached any numbers, and now I'll need to (which will involve doing some estimating). Second, I don't know how much water it takes to put out an average house fire, so I'll need to do some research.

My research on can sizes returned mixed results. Eventually I found out that the standard Campbell's tomato soup can holds 10.75oz, but Wikipedia's chart of standard tin can sizes doesn't have any size with that volume. The nearest one is the No.1 (Picnic) can, so we'll go with that. It has dimensions roughly 6.8 cm by 10.2 cm. The bike in the picture looks like it's a pretty normal size, and most bikes are about 1 m tall. That means the tank has a diameter of 3 m, and since can has a diameter of 6.8 cm, our scale factor is 3*100/6.8, or about 44.1. 10.75oz is 305 mL, so the capacity of the tank is 305*44.1^3 mL, or roughly 26,158 L. Based on the size of the tank, that seems reasonable to me.

The consensus I found online regarding how much water it takes to extinguish a house fire was "it depends on a lot of things", but [this firefighter] estimated that 500 gallons (2273 L) would be enough to put out 85% of house fires. That number seems small to me, and this is a random answer from the internet, but given that our tank holds ten times that amount, I think it's reasonable to say that the water in the tank could put out a house fire. This answer seems even more reasonable when you consider that if the tank didn't hold enough water to put out a fire, the Hornby Island fire authorities would probably build a bigger tank.

This problem was interesting because it tested mathematical reasoning in a way that a lot of math problems don't. It was much vaguer and had less information provided than usual, and it was up to me, the problem solver, to make reasonable assumptions and estimates. I think this is one of the most important mathematical skills in the real world, and we don't focus on it enough in schools.

One time when I was visiting my parents, who live in an apartment building, one of the two elevators had broken down, and I noticed that the wait times for the elevators were much longer - certainly more than double. That got me wondering: what is the relationship between the number of elevators and wait times? In a building with 10 floors, what effect will removing one of the elevators have?

This seems to me like a pretty difficult problem, and in order to make progress one would have to make quite a few assumptions about the behaviour of the elevator-goers and the rate at which they leave/enter the building, but I think it would be a fascinating problem to study, with a huge amount of depth. It could be presented to strong high school students as a long-term inquiry project.