Since raindrops grow as they fall, their surface area increases and therefore the resistance to their falling increases. A raindrop has an initial downward velocity of 10 m/s, and its downward acceleration is $$a = \begin{cases} 9-0.9t & \text{ if } 0 \le t \le 10 \quad \quad \\ 0 & \text{ if } t > 10 \quad \quad \end{cases}$$ If the raindrop is initially 500 meters above the ground, how long does it take (in seconds) to fall?

Time needed = seconds