In fire-service training, there are two ladders of equal height to climb a $20$-foot-high building, one of which has $34$ rungs and the other has $49$ rungs. Two fire service personnel started climbing the building using two ladders. During climbing, they have to shift a fire pipe from one to another. They want the fire pipe to change hands at the lowest distance possible between them. On which rung they should be on their respective ladder to do this?
$9$th step on the first ladder and $13$th step on the second ladder.
On the $k$th step of the first ladder, the distance from the ground is $\frac{20}{34}\cdot k$ feet.
and on the $m$th step of the first ladder, the distance from the ground is $\frac{20}{49}\cdot m$ feet.
The distance between them is $| \frac{20}{34}\cdot k-\frac{20}{49}\cdot m|=\left(\frac{20}{34\cdot 49}\right)|\left(49k-34m\right)|$.
This is minimized if $|\left(49k-34m\right)|=1$, which is true when $k=9$ and $m=13$.
Find the distance of each step from the ground.
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