A set of fireworks mortar shells is launched from the...

A set of fireworks mortar shells is launched from the staging platform at $100 \mathrm{ft} / \mathrm{sec}$ from an initial height of $8 \mathrm{ft}$ above the ground. The height of the fireworks, $h(t),$ can be modeled by $h(t)=-16 t^{2}+100 t+8,$ where $t$ is the time in seconds after launch. a. What is the maximum height that the shells can reach before exploding? b. For how many seconds after launch would the fuses need to be set so that the mortar shells would in fact explode at the maximum height?

A set of fireworks mortar shells is launched from the staging platform at $100 \mathrm{ft} / \mathrm{sec}$ from an initial height of $8 \mathrm{ft}$ above the ground. The height of the fireworks, $h(t),$ can be modeled by
$h(t)=-16 t^{2}+100 t+8,$ where $t$ is the time in seconds after launch.
a. What is the maximum height that the shells can reach before exploding?
b. For how many seconds after launch would the fuses need to be set so that the mortar shells would in fact explode at the maximum height?

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and subject only to the acceleration due to gravity. It is characterized by a constant horizontal component and a vertically accelerated component, which together produce a parabolic trajectory. This concept is fundamental in analyzing the movement of projectiles, including calculating their maximum height and time of flight.

Quadratic Functions

A quadratic function is a second-degree polynomial typically given in the form ax² + bx + c. In the context of projectile motion, the vertical position of an object under gravity can be modeled by a quadratic function. The properties of quadratic functions, such as the symmetry of their graphs (parabolas) and the location of their maximum or minimum values, are crucial in solving problems involving projectile heights.

Vertex of a Parabola

The vertex of a parabola is the point where the function attains its maximum or minimum value, depending on whether it opens downward or upward. For a projectile described by a downward-opening quadratic function, the vertex represents the highest point (maximum height) reached. The time at which this maximum occurs can be calculated using the formula t = -b/(2a), making it a key concept in determining the optimal moment for actions like triggering an explosion at peak altitude.

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