Analyzing Transformations in Periodic Functions for Grade 11

A Ferris wheel has a radius of 15 meters and completes one full rotation every 30 seconds. Write a periodic function to model the height of a passenger above the ground as a function of time. What is the maximum height a passenger can reach?

• 30 meters

• 25 meters

• 35 meters

• 20 meters

A sound wave can be modeled by the function y = 3sin(2πt) + 5, where y is the sound intensity in decibels and t is time in seconds. Analyze the transformations of this function compared to the basic sine function. What is the amplitude and vertical shift?

• Amplitude: 4, Vertical Shift: 1

• Amplitude: 2, Vertical Shift: 0

• Amplitude: 5, Vertical Shift: 3

• Amplitude: 3, Vertical Shift: 5

The tides in a coastal area can be modeled by the function h(t) = 2sin(π/6(t - 12)) + 5, where h is the height of the tide in meters and t is the time in hours. Determine the period of the tide cycle and the maximum height of the tide.

• Period: 6 hours, Maximum height: 5 meters

• Period: 24 hours, Maximum height: 10 meters

• Period: 12 hours, Maximum height: 4 meters

• Period: 12 hours, Maximum height: 7 meters

A pendulum swings back and forth, and its motion can be described by the function θ(t) = 10cos(π/4t), where θ is the angle in degrees and t is time in seconds. What is the period of the pendulum's swing?

• 16 seconds

• 4 seconds

• 8 seconds

• 12 seconds

A car's speedometer shows a periodic fluctuation in speed due to the engine's operation, modeled by the function v(t) = 20cos(2πt) + 60, where v is the speed in km/h and t is time in seconds. What is the range of speeds the car experiences?

• 30 km/h to 70 km/h

• 50 km/h to 90 km/h

• 40 km/h to 80 km/h

• 10 km/h to 50 km/h

A clock's minute hand moves in a circular motion, which can be modeled by the function m(t) = 30sin(π/30t), where m is the angle in degrees and t is time in seconds. How many degrees does the minute hand move in one complete revolution?

• 360 degrees

• 720 degrees

• 180 degrees

• 90 degrees

The temperature in a city varies throughout the day and can be modeled by the function T(t) = 10sin(π/12(t - 6)) + 20, where T is the temperature in degrees Celsius and t is the time in hours. What is the average temperature during the day?

• 25 degrees Celsius

• 30 degrees Celsius

• 20 degrees Celsius

• 15 degrees Celsius

A wave in the ocean can be modeled by the function h(t) = 4sin(2π/5t), where h is the height of the wave in meters and t is time in seconds. Calculate the maximum height of the wave and the period of the wave.

• Maximum height: 4 meters, Period: 5 seconds

• Maximum height: 3 meters, Period: 6 seconds

• Maximum height: 2 meters, Period: 10 seconds

• Maximum height: 5 meters, Period: 4 seconds

A cyclist rides in a circular path with a radius of 10 meters, completing one lap every 20 seconds. Write a periodic function to model the cyclist's distance from the center of the circle as a function of time. What is the distance at t = 10 seconds?

• 5 meters

• 15 meters

• 20 meters

• 10 meters

A seasonal business experiences sales that can be modeled by the function S(t) = 500sin(π/6(t - 3)) + 2000, where S is the sales in dollars and t is the month of the year. What are the peak sales months?

• March, June, September

• August, November, December

• February, May, October

• January, April, July