Worksheet: Modeling with Periodic Functions

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In this worksheet, we will practice modeling real-world situations using periodic functions.

Q1:

A mass attached to the lower end of a spring performs oscillations where ℎ ( 𝑡 ) , the displacement in centimeters of the mass from its equilibrium position, can be modeled by the function

where 𝑡 is measured in seconds.

Find the amplitude, period, and frequency of the displacement.

  • Aamplitude: 8 cm, period: 3 s, frequency: 1 3 Hz
  • Bamplitude: 6 cm, period: 1 8 s, frequency: 8 Hz
  • Camplitude: 8 cm, period: 1 6 s, frequency: 6 Hz
  • Damplitude: 8 cm, period: 1 3 s, frequency: 3 Hz
  • Eamplitude: 4 cm, period: 1 6 s, frequency: 6 Hz

Q2:

A mass attached to the lower end of a spring performs oscillations where ℎ ( 𝑡 ) , the displacement in centimeters of the mass from its equilibrium position, can be modeled by the function

where 𝑡 is measured in seconds.

Find the amplitude, period, and frequency of the displacement.

  • AAmplitude: 5 cm, period: 30 s, frequency: 1 3 0 Hz
  • BAmplitude: 5 cm, period: 1 6 0 s, frequency: 60 Hz
  • CAmplitude: 5 cm, period: 60 s, frequency: 1 6 0 Hz
  • DAmplitude: 5 cm, period: 1 3 0 s, frequency: 30 Hz
  • EAmplitude: 30 cm, period: 1 5 s, frequency: 5 Hz

Q3:

A mass attached to the lower end of a spring performs oscillations where ℎ ( 𝑡 ) , the displacement in centimeters of the mass from its equilibrium position, can be modeled by the function

where 𝑡 is measured in seconds. Find the amplitude, period, and frequency of the displacement.

  • Aamplitude: 4 cm, period: 1 4 s, frequency: 4 Hz
  • Bamplitude: 4 cm, period: 2 s, frequency: 1 2 Hz
  • Camplitude: 4 cm, period: 1 2 s, frequency: 2 Hz
  • Damplitude: 4 cm, period: 4 s, frequency: 1 4 Hz
  • Eamplitude: 2 cm, period: 4 s, frequency: 1 4 Hz

Q4:

A mass attached to the lower end of a spring performs oscillations where ℎ ( 𝑡 ) , the displacement in centimeters of the mass from its equilibrium position, can be modeled by the function where 𝑡 is measured in seconds. Find the amplitude, period, and frequency of the displacement.

  • Aamplitude: 11 cm, period: 6 s, frequency: 1 6 H z
  • Bamplitude: 12 cm, period: 1 1 1 s, frequency: 10 Hz
  • Camplitude: 11 cm, period: 1 1 2 s, frequency: 12 Hz
  • Damplitude: 11 cm, period: 1 6 s, frequency: 6 Hz
  • Eamplitude: 5.5 cm, period: 1 1 2 s, frequency: 12 Hz

Q5:

A Ferris wheel is 20 m in diameter. A ride takes 6 minutes and consists of one complete revolution, starting and finishing at the lowest point. When riders board the Ferris wheel, their seats are 2 m above the ground. How much of a ride is spent more than 13 m above ground?

  • A 4 minutes 24 seconds
  • B 2 minutes 19 seconds
  • C 3 minutes 54 seconds
  • D 2 minutes 49 seconds
  • E 3 minutes 18 seconds

Q6:

In a certain region, monthly precipitation peaks at 24 inches in September and falls to a low of 4 inches in March. Identify the periods when the region is under flood conditions (greater than 22 inches) and drought conditions (less than 5 inches). Give your answer in terms of the nearest day.

  • Afloods: January 23 through April 6, droughts: August 4 through September 25
  • Bfloods: February 4 through March 27, droughts: July 24 through October 7
  • Cfloods: August 4 through September 25, droughts: January 23 through April 6
  • Dfloods: July 24 through October 7, droughts: February 4 through March 27
  • Efloods: October 7 through February 4, droughts: March 27 through July 24

Q7:

A spring is fixed at one end and hangs vertically. Its lower end is pulled 11 cm down from its equilibrium position and released. It performs 8 oscillations every second, and, after 2 seconds, the amplitude of the oscillations is 6 cm. How long does it take for the amplitude of its oscillations to decrease to 0.1 cm? Give your answer to one decimal place.

Q8:

During a 90-day monsoon season, daily rainfall can be modeled by sinusoidal functions. If the rainfall fluctuates between 2 inches on day 10 and 12 inches on day 55, during what period is the rainfall more than 10 inches?

  • Afrom day 9 through day 35
  • Bfrom day 43 through day 66
  • Cfrom day 19 through day 45
  • Dfrom day 41 through day 68
  • Efrom day 64 through day 90

Q9:

The height, ℎ , of a piston can be modeled by the equation ℎ = 2 𝑥 + 6 c o s , where 𝑥 represents the crank angle and ℎ is measured in inches . Find, to 2 decimal places, the height of the piston when the crank angle is 5 5 ∘ .

Q10:

Which of the following is the best model for the temperature fluctuations on a cold winter’s day in a location where the warmest part of the day is around 2 pm and the coldest around 2 am? Let 𝑇 be the temperature in degrees celsius and 𝑡 be the time after midnight in hours.

  • A 𝑇 = 4 𝜋 6 ( 𝑡 + 1 3 ) + 5 s i n
  • B 𝑇 = 3 𝜋 1 2 ( 𝑡 + 7 ) + 3 0 s i n
  • C 𝑇 = 1 0 𝜋 6 ( 𝑡 − 8 ) + 9 s i n
  • D 𝑇 = 3 𝜋 1 2 ( 𝑡 − 1 4 ) + 2 c o s
  • E 𝑇 = 5 𝜋 1 2 ( 𝑡 − 1 5 ) + 3 c o s

Q11:

London is in the northern hemisphere and the number of hours of daylight varies throughout the year. On December 21st they have around 7 hours and 49 minutes of daylight, while on June 21st they have 16 hours and 38 minutes. Which of the following models would best approximate the number of hours of daylight in London on a given day, where ℎ is the number of hours of daylight and 𝑑 is the number of days since the first of January?

  • A ℎ = 5 − 4 4 𝜋 3 6 5 ( 𝑑 − 1 3 ) s i n
  • B ℎ = 2 4 − 3 2 𝜋 3 6 5 ( 𝑑 − 7 ) s i n
  • C ℎ = 9 − 9 4 𝜋 3 6 5 ( 𝑑 − 3 0 ) s i n
  • D ℎ = 1 2 − 4 . 4 2 𝜋 3 6 5 ( 𝑑 + 1 1 ) c o s
  • E ℎ = 3 − 5 2 𝜋 3 6 5 ( 𝑑 + 1 1 ) c o s

Q12:

Shady and his friends went on a trip to the London Eye. The whole trip lasted for 30 minutes. When they got into the pod, they were 15 m above the Thames. Given that the diameter of the London Eye is 120 m, write the equation for the height, , of the pod above the Thames minutes after they got into it.

  • A
  • B
  • C
  • D
  • E

Q13:

The depth of the water in a fishing port is usually 28 metres. The tidal movement is represented by 𝑆 = 4 ( 1 5 𝑛 ) + 2 8 s i n ∘ , where 𝑛 is the time elapsed in hours after midnight. How many times in a day is the depth of the water 24 metres?

  • Athree times
  • Btwo times
  • Cfour times
  • Done time

Q14:

The temperature fluctuation in London over the course of a day can be modeled using a sinusoidal function.

Given that the maximum temperature of 2 1 ∘ C was at 3 pm, and the minimum temperature of 1 0 ∘ C was at 3 am, write an expression for the temperature in terms of 𝑡 , the number of hours after midnight.

  • A 𝑇 = 5 . 5 𝜋 1 2 ( 𝑡 + 1 5 ) + 1 5 . 5 c o s
  • B 𝑇 = 5 . 5 𝜋 1 2 ( 𝑡 − 1 5 ) + 2 1 c o s
  • C 𝑇 = 𝜋 1 2 ( 𝑡 − 1 5 ) + 2 1 c o s
  • D 𝑇 = 5 . 5 𝜋 1 2 ( 𝑡 − 1 5 ) + 1 5 . 5 c o s
  • E 𝑇 = 𝜋 1 2 ( 𝑡 + 1 5 ) + 2 1 c o s

Hence, find the temperature at 7 pm.

  • A 𝑇 = 1 8 . 2 5 ∘ C
  • B 𝑇 = 2 1 . 5 ∘ C
  • C 𝑇 = 1 0 . 7 4 ∘ C
  • D 𝑇 = 2 3 . 7 5 ∘ C
  • E 𝑇 = 2 0 . 1 3 ∘ C

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