#### Answer

t = 1.53 seconds

#### Work Step by Step

The initial height of the roof (and the stone) is 30m. The stone is thrown downward with a speed of 12.0 m/s, therefore the velocity = -12.0 m/s (we designate upwards as positive).
We use the equation:
$x = x0 + v0*t + (1/2)at^2 $
where x0 and v0 are initial position and initial velocity, respectively
The acceleration in this case is that due to gravity, and that is $a = -9.81 m/s^2$ (negative because it is downwards, which we designate as negative)
We are solving for t when x = 0 (when the stone hits the ground), therefore:
$0 = 30m + (-12m/s)*t +(1.2)(-9.81m/s^2)t^2 $
$0 = 30 -12t -4.905t^2 $
We now have a quadratic, which we can either graph to find the roots or use the quadradic equation:
x = $ \frac{-b + \sqrt(b^2 - 4ac) }{2a}$ and x = $ \frac{-b - \sqrt(b^2 - 4ac)
using a = -4.905, b = -12, c = 30
We then get two roots, but we take only the positive one since time can only be positive and we get that t = 1.53 seconds