Let’s go through each part of Task 5 one by one. The task gives us the equation:

$h = -16t^2 + 84t + 100$

where:

• $h$ is the height of the fireworks in feet,
• $t$ is the time in seconds since they were fired.

This equation is a quadratic function that models the height of the fireworks over time.

Part A: How long are the fireworks in the air?

To find how long the fireworks are in the air, we need to determine when they hit the ground, i.e., when $h = 0$.

1. Set $h = 0$: $-16t^2 + 84t + 100 = 0$

2. Solve this quadratic equation using the quadratic formula: $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = -16$, $b = 84$, and $c = 100$.

3. Substitute these values and solve for $t$.

Part B: How high did the fireworks get?

To find the maximum height, we need to determine the vertex of this parabola. The time at which the maximum height occurs is given by:

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

Substitute $a = -16$ and $b = 84$ to find the time at which the fireworks reach their maximum height. Then, substitute this value of $t$ back into the equation for $h$ to get the maximum height.

Part C: How high were the fireworks 1 second after they left the roof?

Substitute $t = 1$ into the equation:

$h = -16(1)^2 + 84(1) + 100$

Calculate this to find the height at $t = 1$ second.

Would you like a detailed step-by-step solution for each part, or have any specific questions about one of them?

Additional Questions to Explore:

1. What would happen to the maximum height if the initial velocity were higher than 84 feet per second?
2. How would the equation change if the building height were different from 100 feet?
3. What does the negative coefficient of $t^2$ in the equation signify in terms of the fireworks' motion?
4. Can we determine the total distance traveled by the fireworks until they reach the ground?
5. What is the significance of the discriminant ($b^2 - 4ac$) in the context of the time the fireworks stay in the air?

Tip:

For quadratic equations describing motion, the vertex represents the peak or maximum height if the parabola opens downward.