69(1003). A cyclist and a motorcyclist are moving towards an intersection along two mutually perpendicular roads. At a certain moment, the cyclist is 8 km away, and the motorcyclist is 15 km away from the intersection. After how many minutes will the distance between them be 5 km, if the cyclist's speed is $\frac{1}{3}$ km/min, and the motorcyclist's speed is 1 km/min?
Solution. Let $OA=8$ km, $OB=15$ km (Fig. 30). If the distance of 5 km between the motorcyclist and the cyclist will be in $t$ minutes $\left(A_{1} B_{1}=5\right.$ km), then $A A_{1}=\frac{1}{3} t$ (km), $B B_{1}=t$ (km). Then $O A_{1}=O A-\frac{1}{3} t=$

![](https://cdn.mathpix.com/cropped/2024_05_21_98932225530737b82071g-134.jpg?height=420&width=557&top_left_y=1937&top_left_x=1161)

Fig. 30
$=8-\frac{1}{3} t, O B_{1}=O B-t=15-t$. By the Pythagorean theorem from

$\triangle O A_{1} B_{1}$ we have:

$$
\left(8-\frac{1}{3} t\right)^{2}+(15-t)^{2}=25
$$

After transformations, we get:

$$
\begin{gathered}
5 t^{2}-159 t+1188=0 \\
t_{1}=12\left(\text { min), }, t_{2}=19.8(\text { min }) .\right.
\end{gathered}
$$

Both roots satisfy the condition of the problem: in the first case ($t=12$ min) the motorcyclist and the cyclist have not yet reached the intersection, in the second case ($t=19.8$ min) the motorcyclist will have crossed the intersection.