Solve each problem. In 1981 Vladimir Poliakov (USSR) set a...

Solve each problem. In 1981 Vladimir Poliakov (USSR) set a world record of $19 \mathrm{ft} \frac{3}{4}$ in. for the pole vault (www.polevault.com). To reach that height, Poliakov obtained a speed of approximately 36 feet per second on the runway. The function $h=-16 t^{2}+36 t$ gives his height $t$ seconds after leaving the ground. a) Use the formula to find the exact values of $t$ for which his height was 18 feet. b) Use the accompanying graph to estimate the value of $t$ for which he was at his maximum height. c) Approximately how long was he in the air?

Solve each problem.
In 1981 Vladimir Poliakov (USSR) set a world record of $19 \mathrm{ft} \frac{3}{4}$ in. for the pole vault (www.polevault.com). To reach that height, Poliakov obtained a speed of approximately 36 feet per second on the runway. The function $h=-16 t^{2}+36 t$ gives his height $t$ seconds after leaving the ground.
a) Use the formula to find the exact values of $t$ for which his height was 18 feet.
b) Use the accompanying graph to estimate the value of $t$ for which he was at his maximum height.
c) Approximately how long was he in the air?

Key Concepts

Projectile Motion

Projectile motion describes the motion of an object launched into the air, subject only to the acceleration due to gravity. The path of the projectile is a parabola, and understanding this concept helps in determining key parameters such as maximum height, time of flight, and the points where the projectile reaches a specific height.

Time of Flight

The total time of flight of a projectile is the duration from launch to landing. It is determined by finding the time at which the height function returns to its initial level (often the ground). This concept is crucial in analyzing the complete trajectory of a projectile under the influence of gravity.

Vertex of a Parabola

The vertex of a parabola, given by the point (?b/(2a), f(?b/(2a))), represents the maximum or minimum value of the quadratic function. In the context of projectile motion, the vertex corresponds to the highest point reached by the object.

Quadratic Formula

The quadratic formula t = (-b ± ?(b² - 4ac)) / (2a) is a method for finding the exact roots of any quadratic equation. It is particularly useful in solving for time or other variables when the quadratic function models a physical situation, such as the height of a projectile at a given time.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(t) = at² + bt + c. They are commonly used to model real-world situations such as projectile motion, where the path of an object under uniform acceleration (like gravity) forms a parabolic curve.

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