A worker is being raised in a bucket lift at a constant speed of [tex]$3 ft / s$[/tex]. When the worker's arms are 10 ft off the ground, her coworker throws a measuring tape toward her. The measuring tape is thrown from a height of 6 ft with an initial vertical velocity of [tex]$15 ft / s$[/tex].
Projectile motion formula:
[tex]$h=-16 t^2+v t+h_0$[/tex]
[tex]$t=$[/tex] time, in seconds, since the measuring tape was thrown
[tex]$h=$[/tex] height, in feet, above the ground
Which system models this situation?
A. [tex]$h=3 t+10$[/tex] and [tex]$h=-16 t^2+15 t+6$[/tex]
B. [tex]$10 t +3$[/tex] and [tex]$h =-16 t ^2+6 t +15$[/tex]
C. [tex]$-16 t^2+3 t+10$[/tex] and [tex]$h=-16 t^2+15 t+6$[/tex]
D. [tex]$-16 t^2+10 t+3$[/tex] and [tex]$h=-16 t^2+6 t+15$[/tex]
Answer (2)
The correct system of equations modeling the situation consists of the worker's height as h = 3 t + 10 and the measuring tape's height as h = − 16 t 2 + 15 t + 6 . Therefore, the answer is option A: h = 3 t + 10 and h = − 16 t 2 + 15 t + 6 .
;
The worker's height is modeled as h = 3 t + 10 .
The measuring tape's height is modeled as h = − 16 t 2 + 15 t + 6 .
The system of equations that models the situation is h = 3 t + 10 and h = − 16 t 2 + 15 t + 6 .
The correct answer is h = 3 t + 10 and h = − 16 t 2 + 15 t + 6 .
Explanation
Problem Analysis Let's analyze the problem. We have a worker rising at a constant speed and a measuring tape thrown upwards. We need to find the equations that model their heights over time.
Worker's Height The worker starts at a height of 10 ft and rises at a constant speed of 3 ft/s. Therefore, the worker's height h at time t is given by the equation: h = 3 t + 10
Measuring Tape's Height The measuring tape is thrown from a height of 6 ft with an initial vertical velocity of 15 ft/s. Using the projectile motion formula h = − 16 t 2 + v t + h 0 , where v is the initial velocity and h 0 is the initial height, the height of the measuring tape at time t is given by the equation: h = − 16 t 2 + 15 t + 6
System of Equations Combining the two equations, the system that models this situation is: h = 3 t + 10 h = − 16 t 2 + 15 t + 6
Final Answer Therefore, the correct answer is: h = 3 t + 10 and h = − 16 t 2 + 15 t + 6
Examples
Understanding projectile motion and linear motion is crucial in many real-world scenarios. For example, when designing amusement park rides, engineers need to calculate the trajectory of moving objects to ensure safety and create an exciting experience. Similarly, in sports, understanding these principles helps athletes optimize their performance, such as a basketball player calculating the optimal angle and velocity to make a shot or a baseball player predicting where a ball will land.
