The Fire-fighter's Car Keys
Problem
First the fire :
If, just as a first approximation, we don't worry about a filled bucket being heavier to carry, what is the best point on the river bank for the fire-fighter to fill the bucket ?.
If you need to do a calculation with lengths, what measurements will you need to make from your diagram ?
It's a general solution you are looking for, so you may need two or three different arrangements or diagrams to see how the solution relates to the positions of the fire-fighter and the fire.
Now the keys :
Draw a horizontal line. Fix two pins at different horizontal levels above the line. A set of keys slides on a string and the string runs over those two pins (the pins are not directly underneath one another). Gradually let out the string length until the weight of the keys brings the string down to touch the drawn horizontal line.
Can you see the connection between this problem and the fire problem above ?
Getting Started
Start with a diagram. What measurements might define the particular arrangement you have drawn ?
From your drawing try some possible positions along the bank. Which gave the shortest overall distance ?
If you are trying to find a relationship between the best bucket-filling position and the positions of the fire and fire-fighter it might be useful to test some particular arrangements.
For example, suppose the fire is half the distance from the river that the fire-fighter is. Try that the other way around as well.
Suppose the fire is still half the fire-fighter's distance but both are now nearer the river than before.
Or further away.
Suppose the fire is still half the fire-fighter's distance but the two positions are further from each other along the bank. Closer along the bank ?
How are you finding the best position each time - by measurement ?How about using a spreadsheet ? You could calculate the total distance for bucket-filling positions all along the bank and see which gives the smallest value. What calculation is needed ?
Do one yourself with a calculator to help you see what calculation (formula) you need to make the spreadsheet do.
For the keys problem set up a real experiment with string, pins and keys.
Student Solutions
The fireman must stop at some point of the riverbank on his way to the fire. If we reflect the fire in the riverbank then our problem is equivalent to finding the shortest path, via the riverbank, to the reflected fire. This is just a straight line, so we reflect this line back in the riverbank to get our actual path. This shortest path is characterised by the fact that itarrives and leaves the riverbank at the same angle. We say that the angle of incidence and the angle of reflection are the same.
Can you solve the second, seemingly unrelated problem, using the first?
Teachers' Resources
See the Hint section for detailed support for students, but the main aims of this problem are :
1. If a calculation approach is adopted, to see the value of using a spreadsheet . Thinking through the calculation required and the spreadsheet commands (formulae) necessary to achieve that are prime Stage 4 activities.
2. To account for the general result that emerges, in such a way that the 'solution' then becomes obvious, adds an additional visualisation to the student's repertoire of possibilities.