Answer
(a) $3.7$
(b) $67m/s$
Work Step by Step
(a) The rain falls vertically while the bus moves horizontally. Thus, the vectors make a triangle. We can find the required ratio as
$\frac{\vec v_{rg}}{\vec v_{bg}}=\frac{\vec v_{rb}cos\theta}{\vec v_{rb}sin\theta}$
Where $\vec v_{rg},\vec v_{bg}$ and $\vec v_{rb}$ represent velocity of rain relative to ground, velocity of bus relative to ground, and velocity of rain relative to bus.
$\implies \frac{\vec v_{rg}}{\vec v_{bg}}=\frac{1}{tan\theta}$
$ \frac{\vec v_{rg}}{\vec v_{bg}}=\frac{1}{tan15^{\circ}}=3.7$
(b) From part (a), we know that:
$\frac{\vec v_{rg}}{\vec v_{bg}}=3.7$
$\implies \vec v_{rg}=3.7\vec v_{bg}$
$\vec v_{rg}=3.7(18m/s)$
$\vec v_{rg}=67m/s$
