A model rocket is launched with an initial velocity of 150 ft / sec from a height of 40 ft. The

step 1 We are given the function in the statement of the problem, that is, ST equal to minus 16 t squar

A model rocket is launched with an initial velocity of 150...

A model rocket is launched with an initial velocity of $150 \mathrm{ft} / \mathrm{sec}$ from a height of 40 ft. The function $s(t)=-16 t^{2}+150 t+40$ gives the height of the rocket, in feet, $t$ seconds after it has been launched. Determine the time at which the rocket reaches its maximum height and find the maximum height.

A model rocket is launched with an initial velocity of $150 \mathrm{ft} / \mathrm{sec}$ from a height of
40 ft. The function $s(t)=-16 t^{2}+150 t+40$ gives the height of the rocket, in feet, $t$ seconds after it has been launched. Determine the time at which the rocket reaches its maximum height and find the maximum height.

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Key Concepts

Vertex of a Parabola

The vertex of a parabola is the point where the function attains its maximum or minimum value. In scenarios involving a downward opening quadratic (where a is negative), the vertex represents the maximum point of the function. The time coordinate of the vertex can be determined using the formula t = -b/(2a), which is a critical tool in solving optimization problems in various contexts.

Projectile Motion

Projectile motion describes the motion of an object that is thrown or projected into the air, subject only to the acceleration due to gravity. The vertical component of projectile motion is often modeled by a quadratic function, where the initial velocity and the effect of gravity determine the path's characteristics, including maximum height and the time to reach it.

Quadratic Functions

A quadratic function is a second-degree polynomial typically expressed in the form s(t) = at² + bt + c. Its graph is a parabola, which can either open upwards or downwards depending on the sign of the coefficient a. These functions are fundamental in modeling scenarios where changes follow a squared time dependence, such as in many motion problems.

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