9.3. A caravan of camels 1 km long is moving uniformly across the desert. A rider rode from the end of the caravan to the beginning and returned to the end of the caravan. During this time, the caravan traveled 1 km. What distance did the rider travel if

![](https://cdn.mathpix.com/cropped/2024_05_06_56a733f440c96ed2b2bfg-1.jpg?height=63&width=674&top_left_y=2467&top_left_x=337)

$\boldsymbol{S o l u t i o n . ~ W e ~ w i l l ~ o b s e r v e ~ t h e ~ m o v e m e n t ~ o f ~ t h e ~ r i d e r ~ R ~ r e l a t i v e ~ t o ~ t h e ~ e n d ~ o f ~ t h e ~ c a r a v a n ~ C ~}$ Let V be the speed of the caravan, and W be the speed of the rider. Initially, R moves away from C at a speed of $\mathrm{W}-\mathrm{V}$, and after some time $\mathrm{t}$, the distance between R and C will be 1 (i.e., R will be at the beginning of the caravan). Then

$$
(\mathrm{W}-\mathrm{V}) t_{1}=1
$$

Next, R approaches C at a speed of $\mathrm{W}+\mathrm{V}$ and after some time $\mathrm{t}$, will be next to C. Then

$$
(\mathrm{W}+\mathrm{V}) t_{2}=1
$$

During the entire time $t_{1}+t_{2}$, point C travels a distance of 1;

$$
\mathrm{V}\left(t_{1}+t_{2}\right)=1
$$

Expressing $t_{1}$ and $t_{2}$ from (1) and (2) and substituting into (3), we get

$$
\mathrm{V}\left(\frac{1}{W-V}+\frac{1}{W+V}\right)=1
$$

From this equation, we can find the ratio $\frac{W}{V}=x$ :

$\mathrm{V}\left(\frac{1}{V x-V}+\frac{1}{V x+V}\right)=1, \quad \frac{1}{x-1}+\frac{1}{x+1}=1, \quad \frac{2 x}{x^{2}-1}=1, \quad x^{2}-2 x-1=0, x=1+\sqrt{2}$.

Multiplying equation (3) by $x=1+\sqrt{2}$, we find $\quad \mathrm{W}\left(t_{1}+t_{2}\right)=1+\sqrt{2}$.

But the quantity $\mathrm{W}\left(t_{1}+t_{2}\right)$ is precisely the distance traveled by the rider.
Answer: $(1+\sqrt{2})$ km.