07 July 2022

A football on the roof

I am always on the lookout for 'real-life' mathematics that is of potential relevance and interest to students but where the mathematics isn't trivial and the context isn't contrived. Too often the scenario is of potential interest but the mathematics is spurious, and doesn't really offer anything in the actual situation that couldn't be done more easily without mathematics. It is not easy to find good examples, but I think this is one that might provide some opportunities to work on topics such as similar triangles and ratio.

Some students lost a football on a flat roof and wanted to know whether the ball had rolled off and fallen down behind the back of the building (i.e., gone forever) or whether it was worth climbing up to retrieve it (Note 1). There weren't any tall buildings nearby that they could access to get a good view of the roof. What they needed to know was how far back from the building they needed to stand so as to be sure that if the ball was there they would be able to see it.

"Can you see it?"
"No, but I just need to go back a bit further."
"If it was there, you'd be able to see it by now."
"I'm not sure."

In particular, going as far back from the building as possible, given the constraints of the surrounding buildings, did the fact that they couldn't see the football mean that it was definitely not there, or could it be that it was just not visible over the edge of the building (Figure 1)?

Figure 1. A football on the roof

This problem is reminiscent of lessons in which students determine the height of a tree near the school using a clinometer, but it feels somehow different. There is not usually any good reason for wanting to know the height of a tree, and it is usually hard to find any way to decide afterwards whether the students' estimates are reasonably accurate or not. In this case, there is a clear 'need to know' and, ultimately, when the site manager brings a ladder, the students would discover if they were right or wrong, so it feels as though something is at stake.

A good way to start would be to decide on simplifying assumptions that it seems sensible to make; i.e., things that we might sensibly choose to ignore. For example:

  • assume that the ground and all roofs are perfectly horizontal
  • assume that the roof in question is free from any debris
  • assume that the building has height 4 m and goes back 6 m
  • assume that the football is perfectly spherical, with diameter 22 cm
  • assume (worst-case scenario) that the ball is right at the back of the roof, against the brick wall

Students may suggest more outlandish things, such as assuming that light travels in straight lines or that the curvature of the earth is negligible, and I would include these as well if they raised them.

Figure 2. Careful analysis (diagram not drawn to scale)

Let's start with a careful analysis, which uses trigonometry and is 'a sledgehammer to crack a nut' for this simple scenario. This is not the approach that I would envisage students taking.

Lots of the work has been done in the diagram (Figure 2), and our units are metre throughout.

We have

$$\tan \theta = \frac{r}{l}$$

$$\tan 2\theta = \frac{h}{d}$$

Using the identity

$$\tan 2\theta \equiv \frac{2 \tan \theta}{1-\tan^2 \theta}$$

we obtain

$$\frac{h}{d}= \frac{2 \left( \frac{r}{l} \right) }{1- \left( \frac{r}{l} \right)^2},$$

giving

$$d= \frac{h(l^2-r^2)}{2rl}.$$

We can now substitute in some reasonable values:

  • $h=4-1.8=2.2$; the height of the building subtract the maximum eye height of the student when standing on tip toes or jumping,
  • $l=6-0.22=5.78$; the depth of the shed subtract the diameter of the football, and
  • $r=0.11$.

This gives $d=57.8$, so the student would just be able to see the top of the football from about 58 metre back from the shed.

But the trigonometry here is overkill for the nature of this problem and the accuracy required, so it would be much quicker and more reasonable to use the simplified diagram shown in Figure 3.

Figure 3. Rougher analysis (diagram not drawn to scale)

For this rougher analysis, we don't need to use $\tan$ explicitly and can just equate corresponding ratios in similar triangles.

So,

$$\frac{d}{h}=\frac{l}{2r},$$

giving that

$$d=\frac{hl}{2r},$$

so, with the same values as above, this again gives $d=57.8$, correct to 1 decimal place, and the same conclusion that the student would just be able to see the top of the football from about 58 metre back from the shed.

In the situation where $l  \gg r$, we can see that in our first equation

$$d= \frac{h(l^2-r^2)}{2rl}$$

the bracket $(l^2-r^2)$ will, to a good approximation, reduce to $l^2$, giving

$$d \approx \frac{hl^2}{2rl}=\frac{hl}{2r},$$

as before. So, all routes lead to an answer of about 58 metre.

But, what if the playground extends only, say, 40 metre before meeting another building? Would it be worth the students going inside and fetching a chair to stand on? Would that make enough difference to be worth the trouble?

The beauty of having derived a formula is that a question like this can be answered instantly by substitution. All that changes here is that $h$ reduces from 2.2 to, say, 1.7.

So,

$$d=\frac{hl}{2r}=\frac{1.7 \times 5.78}{2 \times 0.11}=44.7,$$

and so this would not quite be enough to work within the available space, since $44.7 > 40$. Rearranging the equation to give

$$h=\frac{2rd}{l}$$

reveals that, unless you can find a stool of height at least $2.2 -1.52 = 0.68$ metre, then there is no point bothering.

The mathematics here is not profound, but the result is not guessable without it. I think we need more tasks like this, where a little bit of mathematics (not pages and pages) tells you something practically useful that you couldn't have guesstimated accurately enough without it.

Questions to reflect on

1. Would your students find a task like this credibly realistic and engaging? How might you improve it?

2. What 'real-life' tasks do you use that are both non-trivial mathematically and non-embarrassing in terms of correspondence with reality?

Note

1. Disclaimer: Nothing in this post should be taken to endorse climbing onto roofs to retrieve footballs!

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