The staging platform for a fireworks display is 6 ft above...

The staging platform for a fireworks display is $6 \mathrm{ft}$ above ground, and the mortars leave the platform at $96 \mathrm{ft} / \mathrm{sec}$. The height of the mortars $h(t)$ (in feet) can be modeled by $h(t)=-16(t-3)^{2}+150,$ where $t$ is the time in seconds after launch. a. If the fuses are set for 3 sec after launch, at what height will the fireworks explode? b. Will the fireworks explode at their maximum height? Explain.

The staging platform for a fireworks display is $6 \mathrm{ft}$ above ground, and the mortars leave the platform at $96 \mathrm{ft} / \mathrm{sec}$. The height of the mortars $h(t)$ (in feet) can be modeled by $h(t)=-16(t-3)^{2}+150,$ where $t$ is the time in seconds after launch.
a. If the fuses are set for 3 sec after launch, at what height will the fireworks explode?
b. Will the fireworks explode at their maximum height? Explain.

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Key Concepts

Quadratic Functions and Vertex Form

Quadratic functions are used to model parabolic paths, and the vertex form of a quadratic equation provides direct information about the maximum or minimum point of the graph. In contexts such as projectile motion, the vertex represents the highest point reached, which is critical for determining the time and height at which an object peaks in its trajectory.

Time Shifts in Function Models

Time shifts in function models occur when the variable in the function is adjusted by a constant, such as (t - 3) in the quadratic formula. This shift represents a delay or advancement in time, which is essential for accurately modeling events that do not start at time zero and for aligning the mathematical model with real-world timing, such as when a fuse is set to ignite after a specific time delay.

Projectile Motion

Projectile motion involves the movement of an object under the influence of gravity, where its path can be modeled by a parabolic curve. This concept explains how objects thrown or launched into the air follow trajectories that can be predicted using physics and mathematical equations, accommodating factors such as initial velocity, launch angle, and height.

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