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.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, commonly expressed in the form ax² + bx + c. They produce parabolic graphs which, depending on the sign of the coefficient a, open upward or downward. In physical applications such as projectile motion, quadratic functions model the object's height over time with the vertex representing the extremum—either a maximum or a minimum point.

Vertex of a Parabola

The vertex of a parabola is the point at which the function attains its maximum or minimum value. For a quadratic function modeling projectile motion (where a is negative and the parabola opens downward), the vertex represents the maximum height reached by the projectile. The vertex can be determined using the formula -b/(2a) for the time coordinate and then evaluating the function at that time to find the maximum height.

Projectile Motion

Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. It is typically modeled using quadratic functions, which incorporate initial velocity, launch angle, and gravitational acceleration to determine the object's trajectory, including its maximum height, range, and time of flight.

Transcript

Please give Ace some feedback