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'The Fire-fighter's Car Keys' printed from https://nrich.maths.org/
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.