Questions: Which situations are modeled accurately by a periodic 22 function? Select all that apply. the height of a ball tossed in the air as a function of time the vertical height of a point on a rotating wheel as a function of time the height of a swinging pendulum as a function of time the distance from the earth to the moon as a function of time

Solution Steps

To determine which situations are modeled accurately by a periodic function, we need to identify scenarios where the behavior repeats at regular intervals. A periodic function is one that repeats its values in regular intervals or periods.

1. The height of a ball tossed in the air as a function of time is not periodic because the ball follows a parabolic trajectory and does not repeat its motion in a regular cycle.
2. The vertical height of a point on a rotating wheel as a function of time is periodic because the wheel rotates in a circle, repeating its position at regular intervals.
3. The height of a swinging pendulum as a function of time is periodic because the pendulum swings back and forth in a regular, repeating cycle.
4. The distance from the earth to the moon as a function of time is not strictly periodic because it varies due to the elliptical orbit and other gravitational influences, though it may have periodic components.

To determine which situations are modeled accurately by a periodic function, we analyze each scenario for regularity in their behavior over time.

1. Height of a ball tossed in the air: This follows a parabolic trajectory, described by \( h(t) = -gt^2 + v_0t + h_0 \), where \( g \) is the acceleration due to gravity, \( v_0 \) is the initial velocity, and \( h_0 \) is the initial height. This function is not periodic.

2. Vertical height of a point on a rotating wheel: This can be modeled by a sine function, \( h(t) = r \sin(\omega t + \phi) \), where \( r \) is the radius, \( \omega \) is the angular velocity, and \( \phi \) is the phase shift. This function is periodic with period \( T = \frac{2\pi}{\omega} \).

3. Height of a swinging pendulum: The height can also be modeled by a sine function, similar to the rotating wheel, \( h(t) = A \sin(\omega t + \phi) \), where \( A \) is the amplitude. This function is periodic with period \( T = \frac{2\pi}{\omega} \).

4. Distance from the Earth to the Moon: This distance varies due to the elliptical orbit and is not periodic in a strict sense, as it does not repeat in a regular cycle.

From the analysis, the situations that are periodic are:

• The vertical height of a point on a rotating wheel as a function of time.
• The height of a swinging pendulum as a function of time.

Final Answer