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Hank's Gold Mine has a very long vertical shaft
of height $l$.
A light chain of length $l$
passes over a 
small smooth
light fixed pulley at the top of the shaft.
To one end of the chain is attached a bucket $A$ of
negligible mass and to the other a bucket $B$ of mass
$m$. The system is used to raise ore from the mine as follows.
When bucket $A$ is at the top it is filled with mass $2m$
of water and bucket $B$ is filled with mass $\lambda m$
of ore, where $0<\lambda<1$. The buckets are then released,
so that bucket $A$ descends and bucket $B$ ascends. 
When bucket $B$
reaches the top both buckets are emptied and released,
so that bucket $B$ descends and bucket $A$ ascends. The time to
fill and empty the buckets is negligible. Find the
time taken from the moment bucket $A$ is released at the top
until the first time it reaches the top again.
This process
goes on for a very long time. Show that, if the greatest
amount of ore is to be raised in that time, then
$\lambda$ must satisfy the condition $\mathrm{f}'(\lambda)=0$
where
\[\mathrm{f}(\lambda)=\frac{\lambda(1-\lambda)^{1/2}}
{(1-\lambda)^{1/2}+(3+\lambda)^{1/2}}.\]

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