The equation given is:

$h(t) = -0.021t^2 + 0.504t + 98.1$

This is a quadratic function that models a parabola opening downwards (since the coefficient of $t^2$ is negative). The maximum value of this function occurs at the vertex of the parabola.

To find the time $t$ at which the maximum occurs, we use the vertex formula:

$t_{\text{max}} = \frac{-b}{2a}$

Here, $a = -0.021$ and $b = 0.504$.

Let's calculate $t_{\text{max}}$:

$t_{\text{max}} = \frac{-0.504}{2 \times (-0.021)} = \frac{-0.504}{-0.042}$

Let me calculate that for you.The maximum value occurs at $t = 12$ seconds.

Now, to find the maximum height at this time, we substitute $t = 12$ back into the original equation:

$h(12) = -0.021(12)^2 + 0.504(12) + 98.1$

I'll calculate this for you.