A projectile is thrown upward so that its distance above the ground after t seconds is h=-16t+384t. After how many seconds does it reach its max height

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!

A projectile is thrown upward so that its distance above the ground after t seconds is h=-16t+384t. After how many seconds does it reach its max height?

Discover how to calculate the time at which a projectile reaches its maximum height using the quadratic equation h(t) = -16t^2 + 384t. This detailed explanation covers quadratic functions, the vertex theorem, and the key formula for finding maximum or minimum values. Ideal for students in Grades 9-11.

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