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'One and Three' printed from https://nrich.maths.org/
Two motorboats are travelling up and down a lake at constant
speeds and turning at each end without slowing down. They leave
opposite ends A and B at the same instant, passing each other, for
the first time 600 metres from A and, for the second time, 400
metres from B. How long is the lake (the distance from A to B) and
what is the ratio of the speeds of the boats?
We give the solution by Nisha Doshi , Year 9,
The Mount School, York. Well done Nisha this is a really good
solution.
Call the boats $A\prime$ and $B\prime$. The first time they meet,
the distances gone are: 600 metres by $A\prime$ and $(x + 400)$
metres by $B\prime$.
The second time the distances are: $(x + 400 + 400)$ metres by
$A\prime$ and $(600 + 600 + x)$ metres by $B\prime$.
Since the two speeds are constant, the ratios of the distances
travelled each time must be constant, so $$ \frac{600}{x+400} =
\frac{x+800}{x+1200} $$ which leads to
\begin{eqnarray} 600(x + 1200) &=& (x +
400)(x + 800) \\ 600x + 720000 &=& x^2 + 1200x + 320000 \\
x^2 + 600x - 400000 &=& 0 \\ (x - 400)(x + 1000)
&=& 0 \\ \end{eqnarray}
It follows that $x = 400$ metres and that the length of the lake is
$600 + 400 + 400 = 1400$ metres.
The ratio of the speeds must be 600 : 800 or 3 : 4.
FOOTNOTE: You can also use the fact that at the first meeting the
combined distance travelled is one length of the lake and at the
second it is 3 lengths. It follows that if $A\prime$ travels 600
metres by the first meeting it will have travelled three times that
distance by the second meeting, that is 1800 metres. This gives $x
= 1800 - 600 - 400 - 400 = 400$ and hence the length of the lake is
1400 metres.