To determine when a projectile reaches its maximum height, we need to find the time $t$ at which the height function reaches its maximum. The height function is given by:

$h(t) = -16t^2 + 384t$

This is a quadratic function of the form $h(t) = at^2 + bt + c$, where $a = -16$, $b = 384$, and $c = 0$. For a quadratic function, the maximum or minimum value occurs at $t = -\frac{b}{2a}$.

Step-by-step Solution

1. Identify the coefficients: $a = -16, \quad b = 384$

2. Use the formula for the vertex of a parabola: $t = -\frac{b}{2a}$

3. Substitute the values of $a$ and $b$ into the formula: $t = -\frac{384}{2 \times (-16)}$

4. Simplify the expression: $t = \frac{384}{32} = 12$

Conclusion

The projectile reaches its maximum height after 12 seconds.

Would you like more details on the solution or have any questions?

Related Questions:

1. What is the maximum height reached by the projectile?
2. How long will it take for the projectile to hit the ground?
3. What is the velocity of the projectile at its maximum height?
4. How would the maximum height change if the initial velocity were different?
5. How can we find the time when the projectile is at half of its maximum height?

Tip:

For any quadratic equation, finding the vertex using $t = -\frac{b}{2a}$ is a key step to determine maximum or minimum points!