To find the time it takes for the boulder to reach its maximum height, we need to consider the quadratic function h = -5t^2 + 92t + 16. The maximum height is achieved when the boulder reaches the vertex of the parabolic curve.
The formula for the x-coordinate of the vertex of a quadratic function in the form ax^2 + bx + c is given by x = -b/2a.
In this case, a = -5 and b = 92. Plugging these values into the formula, we have x = -92/(2*(-5)) = -92/(-10) = 9.2 seconds. So, it takes 9.2 seconds for the boulder to reach its maximum height.
To find the maximum height, we need to substitute the value of t = 9.2 into the equation for h. Plugging t = 9.2 into the equation h = -5t^2 + 92t + 16, we get h = -5*(9.2)^2 + 92*(9.2) + 16.
Calculating this expression, we have h = -5*(84.64) + 846.4 + 16 = -423.2 + 846.4 + 16 = 439.2.
Therefore, the boulder's maximum height is 439.2 meters.
So, none of the given options for the maximum height and time to reach the maximum height are correct. The correct answers are:
It takes 9.2 seconds to reach the maximum height.
The boulder's maximum height is 439.2 meters.