The height $h(t)$ of the rocket is given by the quadratic function:

$h(t) = -16t^2 + 160t + 6$

To find the time it takes for the rocket to reach its maximum height, we need to determine the vertex of the parabola. For a quadratic function $h(t) = at^2 + bt + c$, the time $t$ at which the maximum or minimum occurs is given by:

$t = \frac{-b}{2a}$

In this case, $a = -16$, $b = 160$, and $c = 6$. Plugging in these values:

$t = \frac{-160}{2(-16)} = \frac{-160}{-32} = 5 \text{ seconds}$

So, the rocket reaches its maximum height 5 seconds after launch.

Finding the Maximum Height:

To find the maximum height, substitute $t = 5$ back into the height function:

$h(5) = -16(5)^2 + 160(5) + 6$

$h(5) = -16(25) + 160(5) + 6$

$h(5) = -400 + 800 + 6 = 406 \text{ feet}$

Conclusion:

• The rocket reaches its maximum height at $t = 5$ seconds.
• The maximum height is 406 feet.

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

Related Questions:

1. How long will it take for the rocket to hit the ground?
2. What is the total time the rocket spends in the air?
3. What is the rocket's velocity when it reaches the maximum height?
4. How would the maximum height change if the initial velocity were 200 feet per second?
5. How high is the rocket after 3 seconds?

Tip:

To solve quadratic functions, the vertex formula is key for determining maximum or minimum points!