This activity will help you meet these educational goals: You will create a function to model a fireworks show and examine the attributes of the function.
You’re in charge of planning a fireworks show. The company you hire proposes using fireworks called mortar fireworks. These fireworks are placed in a tube that sits on the ground or a flat surface and are shot from the tube with an initial velocity that propels them into the sky.
Mortar fireworks have two fuses that are lit at the same time. The first fuse burns fastest and causes the initial force that launches the firework into the sky. The second fuse takes longer to burn. When the second fuse reaches the middle of the firework, the firework explodes and we see the light show in the sky. This second fuse does not add any extra propulsion to the firework while it’s in the air.
You plan to have the company light the fireworks from the ground. Based on information provided by the company, you’ve determined that the fireworks will have an initial velocity of 192 feet/second.
The formula for the vertical motion of an object is h = -16t2 + v0t + h0, where h is the height of the object, h0 is the initial, or starting, height, v0 is the initial velocity, and t is the time in seconds.

Part A
Create a function to model the height of a firework when shot in the air. Explain whether the function will have a maximum or a minimum value.

Part B
Using the equation representing the height of the firework (h = -16t2 + v0t + h0), algebraically determine the extreme value of f(t) by completing the square and finding the vertex. Interpret what the value represents in this situation.

Part C
Graph the quadratic function that represents the height of the firework. Determine an appropriate domain and range for this function, and use those as x and y minimums and maximums.

Part D
Determine the zeros of the function from the graph or algebraically (by factoring). Explain what these zeros mean in this situation.

Part E
Type the correct answer in the box.
Determine the average rate of change for the quadratic function representing the height of the firework with regard to time over the interval [0,4].

Part F
What will the height of the firework be 3 seconds after the launch? How many seconds after launch will it take for the firework to fall to the same height again?

Part G
For the safety of the audience, the fireworks should be set to explode at least 450 feet above the ground. The length of the second fuse in a mortar firework controls when the firework explodes. The fireworks company has these four fuses available:

Fuse A: The firework will explode in 1 to 3 seconds.
Fuse B: The firework will explode in 3 to 5 seconds.
Fuse C: The firework will explode in 6 to 8 seconds.
Fuse D: The firework will explode in 8 to 10 seconds.
Which fuse should the company use for your fireworks display, and why?