Modeling and Graphing Periodic Functions in Real Life

A Ferris wheel has a radius of 15 meters and takes 2 minutes to make one complete revolution. Write a function that models the height of a passenger above the ground as a function of time. Identify the periodic function and its period.

• h(t) = 15 * sin(2πt) + 15; Period = 1 minute

• h(t) = 15 * sin(πt) + 15; Period = 2 minutes

• h(t) = 15 * cos(πt) + 30; Period = 4 minutes

• h(t) = 30 * sin(πt) + 15; Period = 1 minute

The temperature in a city varies throughout the day in a sinusoidal pattern, reaching a maximum of 30°C at noon and a minimum of 10°C at midnight. Write a function to represent the temperature as a function of time and graph it over a 24-hour period.

• T(t) = 20 + 10 * sin((π/12)(t - 6))

• T(t) = 10 + 20 * sin((π/24)(t))

• T(t) = 25 + 5 * cos((π/12)(t - 6))

• T(t) = 15 + 5 * sin((π/6)(t - 12))

A pendulum swings back and forth, completing one full swing every 4 seconds. If the maximum angle of displacement is 30 degrees, write a function to model the angle of the pendulum over time. Identify the periodic function and its characteristics.

• angle(t) = 15 * cos(π * t)

• angle(t) = 30 * cos(π/2 * t)

• angle(t) = 30 * sin(2π * t)

• angle(t) = 30 * sin(π/4 * t)

A sound wave can be modeled by the function y = 3sin(2πt) where y is the displacement and t is time in seconds. Graph this function and identify its amplitude, period, and frequency.

• Amplitude: 3, Period: 1 second, Frequency: 1 Hz

• Amplitude: 4, Period: 0.25 seconds, Frequency: 4 Hz

• Amplitude: 1, Period: 2 seconds, Frequency: 0.5 Hz

• Amplitude: 2, Period: 0.5 seconds, Frequency: 2 Hz

The tides in a coastal area can be modeled by a sine function with a period of 12 hours. If the high tide occurs at 6 AM with a height of 2 meters, write the function that models the tide height over time and graph it.

• h(t) = 2 * sin(π/6(t - 6))

• h(t) = 2 * sin(π/12(t - 6))

• h(t) = 1 * sin(π/6(t + 6))

• h(t) = 3 * sin(π/6(t - 12))

A cyclist rides in a circular path with a radius of 10 meters, completing one lap every 30 seconds. Write a function to model the cyclist's position on the circular path as a function of time. Identify the periodic function and its period.

• (x(t), y(t)) = (20 * cos((2π/30)t), 20 * sin((2π/30)t)); Period = 30 seconds

• (x(t), y(t)) = (5 * cos((2π/15)t), 5 * sin((2π/15)t)); Period = 15 seconds

• (x(t), y(t)) = (10 * cos((2π/60)t), 10 * sin((2π/60)t)); Period = 60 seconds

• (x(t), y(t)) = (10 * cos((2π/30)t), 10 * sin((2π/30)t)); Period = 30 seconds

A light bulb flickers on and off every 0.5 seconds. Write a function to represent the brightness of the bulb over time and graph it. Identify the periodic nature of the function and its characteristics.

• B(t) = 1 for 0 <= t % 0.5 < 0.5; B(t) = 0 for 0.5 <= t % 0.5 < 1, period = 1 second.

• B(t) = 1 for 0 <= t < 1; B(t) = 0 for t >= 1, period = 1 second.

• B(t) = 1 for 0 <= t % 0.5 < 0.25; B(t) = 0 for 0.25 <= t % 0.5 < 0.5, period = 0.5 seconds.

• B(t) = 1 for 0 <= t % 1 < 0.5; B(t) = 0 for 0.5 <= t % 1 < 1, period = 1 second.

The population of a certain species of fish in a lake follows a periodic pattern due to seasonal changes, peaking every 6 months. If the population reaches a maximum of 500 fish, write a function to model this population and graph it over a year.

• P(t) = 400 * sin(πt) + 100

• P(t) = 300 * cos(2πt) + 200

• P(t) = 500 * sin(πt) + 500

• P(t) = 500 * sin(2πt) - 500

A clock's minute hand moves in a circular motion, completing one full rotation every 60 minutes. Write a function to model the angle of the minute hand as a function of time and identify the periodic function and its period.

• A(t) = 3t; Period = 20 minutes

• A(t) = 90t; Period = 120 minutes

• A(t) = 12t; Period = 30 minutes

• A(t) = 6t; Period = 60 minutes

A roller coaster's height above the ground can be modeled by a cosine function with a maximum height of 50 meters and a minimum height of 5 meters, completing a cycle every 20 seconds. Write the function and graph it, identifying its periodic characteristics.

• h(t) = 27.5 + 22.5 * cos(π/10 * t) with a period of 20 seconds.

• h(t) = 30 + 20 * cos(π/20 * t) with a period of 30 seconds.

• h(t) = 40 + 10 * cos(π/5 * t) with a period of 10 seconds.

• h(t) = 25 + 25 * sin(π/10 * t) with a period of 15 seconds.