Real-World Applications of Sine and Cosine Functions

A Ferris wheel has a radius of 20 meters and completes one full rotation every 30 seconds. Write the sine function that models the height of a passenger above the ground as a function of time, and graph it for one full rotation.

• h(t) = 20 * sin(π/15 * t - π/2) + 40

• h(t) = 20 * cos(π/15 * t + π/2) + 40

• h(t) = 20 * sin(π/30 * t) + 20

• h(t) = 20 * sin(π/15 * t) + 60

A sound wave can be modeled by the function y = 3sin(2πft), where f is the frequency in hertz. If the frequency of the sound wave is 5 Hz, graph the function for one period and describe its amplitude and period.

• Amplitude: 3, Period: 0.2 seconds

• Amplitude: 2, Period: 0.4 seconds

• Amplitude: 5, Period: 0.1 seconds

• Amplitude: 3, Period: 0.5 seconds

A lighthouse's beam of light sweeps out an angle of 60 degrees every 10 seconds. If the distance from the lighthouse to a boat is 100 meters, use the cosine function to model the distance of the boat from the lighthouse over time and graph it.

• d(t) = 100 * cos(12t)

• d(t) = 100 * sin(6t)

• d(t) = 100 * cos(6t)

• d(t) = 50 * cos(3t)

A pendulum swings back and forth, and its motion can be modeled by the function y = 4cos(πt/2), where t is time in seconds. Graph the function and determine the maximum height of the pendulum from its resting position.

• 2

• 4

• 8

• 6

A roller coaster follows a path that can be modeled by the function y = 5sin(πx/10), where x is the horizontal distance in meters. Graph the function and identify the points where the coaster is at its highest and lowest.

• Highest point: (0, 5), Lowest point: (10, -5)

• Highest point: (5, 5), Lowest point: (15, -5)

• Highest point: (10, 5), Lowest point: (20, -5)

• Highest point: (5, 10), Lowest point: (15, -10)

A wave in the ocean can be modeled by the function y = 2sin(0.5x), where x is the distance from the shore in meters. Graph the function and determine the distance from the shore where the wave reaches its maximum height.

• 6.28 meters

• 1.57 meters

• 4.71 meters

• 3.14 meters

A tire rotates and its motion can be modeled by the function y = 10cos(θ), where θ is the angle in radians. If the tire completes one full rotation every 2 seconds, graph the function and find the height of a point on the tire after 1 second.

• 5

• 10

• -10

• 0

A swing moves back and forth in a periodic motion that can be modeled by the function y = 3sin(2πt/5), where t is time in seconds. Graph the function and determine how long it takes for the swing to return to its starting position.

• 5 seconds

• 15 seconds

• 2 seconds

• 10 seconds

A sound wave has a frequency of 2 Hz and can be modeled by the function y = 1.5sin(4πt). Graph the function and identify the time intervals when the wave is at its maximum and minimum values.

• Maximum values occur at t = 0.1, 0.6, 1.1, ...; Minimum values occur at t = 0.4, 0.9, 1.4, ...

• Maximum values occur at t = 0.2, 0.7, 1.2, ...; Minimum values occur at t = 0.6, 1.1, 1.6, ...

• Maximum values occur at t = 0.125, 0.625, 1.125, ...; Minimum values occur at t = 0.375, 0.875, 1.375, ...

• Maximum values occur at t = 0.25, 0.75, 1.25, ...; Minimum values occur at t = 0.5, 1.0, 1.5, ...

A satellite dish is designed to reflect signals to a focal point. The dish can be modeled by the function y = 2cos(x), where x is the distance from the center of the dish. Graph the function and explain how the shape of the dish affects signal reception.

• The dish shape absorbs signals, reducing reception quality.

• The shape of the dish, modeled by y = 2cos(x), focuses signals at a focal point, improving signal reception.

• The function y = 2sin(x) models the dish's shape for better reception.

• The dish's flat surface allows signals to scatter evenly.