Worksheet: Modeling with Trigonometric Functions

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 β„Ž(𝑑), the displacement in centimeters of the mass from its equilibrium position, can be modeled by the function β„Ž(𝑑)=8(6πœ‹π‘‘),sin

where 𝑑 is measured in seconds.

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

  • Aamplitude: 8 cm, period: 3 s, frequency: 13 Hz
  • Bamplitude: 6 cm, period: 18 s, frequency: 8 Hz
  • Camplitude: 8 cm, period: 16 s, frequency: 6 Hz
  • Damplitude: 4 cm, period: 16 s, frequency: 6 Hz
  • Eamplitude: 8 cm, period: 13 s, frequency: 3 Hz

Q2:

A mass attached to the lower end of a spring performs oscillations where β„Ž(𝑑), the displacement in centimeters of the mass from its equilibrium position, can be modeled by the function β„Ž(𝑑)=βˆ’5(60πœ‹π‘‘),cos

where 𝑑 is measured in seconds.

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

  • AAmplitude: 5 cm, period: 30 s, frequency: 130 Hz
  • BAmplitude: 5 cm, period: 60 s, frequency: 160 Hz
  • CAmplitude: 30 cm, period: 15 s, frequency: 5 Hz
  • DAmplitude: 5 cm, period: 130 s, frequency: 30 Hz
  • EAmplitude: 5 cm, period: 160 s, frequency: 60 Hz

Q3:

A mass attached to the lower end of a spring performs oscillations where β„Ž(𝑑), the displacement in centimeters of the mass from its equilibrium position, can be modeled by the function β„Ž(𝑑)=4ο€»πœ‹2𝑑,cos

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

  • Aamplitude: 4 cm, period: 4 s, frequency: 14 Hz
  • Bamplitude: 2 cm, period: 4 s, frequency: 14 Hz
  • Camplitude: 4 cm, period: 2 s, frequency: 12 Hz
  • Damplitude: 4 cm, period: 12 s, frequency: 2 Hz
  • Eamplitude: 4 cm, period: 14 s, frequency: 4 Hz

Q4:

A mass attached to the lower end of a spring performs oscillations where β„Ž(𝑑), the displacement in centimeters of the mass from its equilibrium position, can be modeled by the function β„Ž(𝑑)=11(12πœ‹π‘‘),sin where 𝑑 is measured in seconds. Find the amplitude, period, and frequency of the displacement.

  • Aamplitude: 12 cm, period: 111 s, frequency: 10 Hz
  • Bamplitude: 11 cm, period: 6 s, frequency: 16 Hz
  • Camplitude: 11 cm, period: 112 s, frequency: 12 Hz
  • Damplitude: 5.5 cm, period: 112 s, frequency: 12 Hz
  • Eamplitude: 11 cm, period: 16 s, frequency: 6 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?

  • A3 minutes 54 seconds
  • B4 minutes 24 seconds
  • C3 minutes 18 seconds
  • D2 minutes 49 seconds
  • E2 minutes 19 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: August 4 through September 25, droughts: January 23 through April 6
  • Bfloods: January 23 through April 6, droughts: August 4 through September 25
  • Cfloods: October 7 through February 4, droughts: March 27 through July 24
  • Dfloods: February 4 through March 27, droughts: July 24 through October 7
  • Efloods: July 24 through October 7, droughts: February 4 through March 27

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

Q9:

The height, β„Ž, of a piston can be modeled by the equation β„Ž=2π‘₯+6cos, where π‘₯ represents the crank angle and β„Ž is measured in inches . Find, to 2 decimal places, the height of the piston when the crank angle is 55∘.

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 𝑑 be the time after midnight in hours.

  • A 𝑇 = 5 ο€» πœ‹ 1 2 ( 𝑑 βˆ’ 1 5 )  + 3 c o s
  • B 𝑇 = 3 ο€» πœ‹ 1 2 ( 𝑑 + 7 )  + 3 0 s i n
  • C 𝑇 = 1 0 ο€» πœ‹ 6 ( 𝑑 βˆ’ 8 )  + 9 s i n
  • D 𝑇 = 4 ο€» πœ‹ 6 ( 𝑑 + 1 3 )  + 5 s i n
  • E 𝑇 = 3 ο€» πœ‹ 1 2 ( 𝑑 βˆ’ 1 4 )  + 2 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 β„Ž is the number of hours of daylight and 𝑑 is the number of days since the first of January?

  • A β„Ž = 3 βˆ’ 5 ο€Ό 2 πœ‹ 3 6 5 ( 𝑑 + 1 1 )  c o s
  • B β„Ž = 1 2 βˆ’ 4 . 4 ο€Ό 2 πœ‹ 3 6 5 ( 𝑑 + 1 1 )  c o s
  • C β„Ž = 2 4 βˆ’ 3 ο€Ό 2 πœ‹ 3 6 5 ( 𝑑 βˆ’ 7 )  s i n
  • D β„Ž = 9 βˆ’ 9 ο€Ό 4 πœ‹ 3 6 5 ( 𝑑 βˆ’ 3 0 )  s i n
  • E β„Ž = 5 βˆ’ 4 ο€Ό 4 πœ‹ 3 6 5 ( 𝑑 βˆ’ 1 3 )  s i n

Q12:

Anthony 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, β„Ž, of the pod above the Thames 𝑑 minutes after they got into it.

  • A β„Ž = 7 5 βˆ’ 6 0 ο€Ό πœ‹ 𝑑 1 5  c o s
  • B β„Ž = 7 5 βˆ’ 6 0 ο€Ό πœ‹ 𝑑 3 0  c o s
  • C β„Ž = 7 5 + 6 0 ο€Ό πœ‹ 𝑑 1 5  c o s
  • D β„Ž = 1 5 + ο€Ό πœ‹ 𝑑 1 5  c o s
  • E β„Ž = 1 5 βˆ’ ο€Ό πœ‹ 𝑑 1 5  c o s

Q13:

The depth of the water in a fishing port, 𝑆, is affected by the tides. It can be represented by 𝑆=4(15𝑛)+28sin∘, where 𝑆 is measured in meters and 𝑛 is the time elapsed, in hours, after midnight. How many times a day is the depth of the water exactly 24 meters?

  • AFour times
  • BOne time
  • CTwo times
  • DThree times

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 21∘C was at 3 pm, and the minimum temperature of 10∘C was at 3 am, write an expression for the temperature in terms of 𝑑, the number of hours after midnight.

  • A 𝑇 = 5 . 5 ο€» πœ‹ 1 2 ( 𝑑 βˆ’ 1 5 )  + 1 5 . 5 c o s
  • B 𝑇 = 5 . 5 ο€» πœ‹ 1 2 ( 𝑑 + 1 5 )  + 1 5 . 5 c o s
  • C 𝑇 = ο€» πœ‹ 1 2 ( 𝑑 βˆ’ 1 5 )  + 2 1 c o s
  • D 𝑇 = ο€» πœ‹ 1 2 ( 𝑑 + 1 5 )  + 2 1 c o s
  • E 𝑇 = 5 . 5 ο€» πœ‹ 1 2 ( 𝑑 βˆ’ 1 5 )  + 2 1 c o s

Hence, find the temperature at 7 pm.

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

Q15:

Outside temperatures over 24 hours can be modeled as a sinusoidal function. On a day with an average temperature of 70∘F, the maximum temperature of 84∘F is recorded at 6 pm. Find, to the nearest degree, the temperature at 7 am.

Q16:

In a certain location, the temperature over the course of a day varies between a minimum of 64∘F at 6 am and a maximum of 86∘F. If the temperature is modeled by a sinusoidal function, what is the first time in the day when the temperature is 70∘F?

  • A1:49 am
  • B4:12 am
  • C11:49 pm
  • D7:49 am
  • E10:12 am

Q17:

The percentage grade of a road is defined as the change in height of the road over a 100-foot horizontal distance. For example, if the road rises 5 feet over a horizontal distance of 100 feet, it will have a 5% grade.

What is the percentage grade of a road that makes an angle of 4∘ with the horizontal?

Q18:

A spring is fixed at one end and hangs vertically. Its lower end is pulled 10 cm down from its equilibrium position and released. It performs 22 oscillations every second, and, after 3 seconds, the amplitude of the oscillations is 2 cm. How long does it take for the amplitude of its oscillations to decrease to 0.1 cm? Give your answer to three decimal places.

Q19:

Michael and Hannah went swimming in the sea at 2 pm, when it was high tide. The change in the height of water with respect to the annual average is given by β„Ž=5ο€Ό4πœ‹25π‘‘οˆcos, where 𝑑 is the time, in hours, after any high tide.

At what time was the next high tide?

  • A2:30 pm
  • B12:00 am
  • C2:00 am
  • D2:00 pm
  • E2:30 am

When will it be high tide in the afternoon three days later?

  • A5 pm
  • B12 am
  • C12 pm
  • D5 am
  • E5:30 pm

They want to go back to the same beach three days later, in the afternoon, and would like the height of the water to be at least 4 meters above the annual height average. Between what times should they go?

  • Abetween 6:15 pm and 4:12 am
  • Bbetween 1:17 pm and 11:12 pm
  • Cbetween 3:17 pm and 1:12 am
  • Dbetween 3:42 pm and 6:17 pm
  • Ebetween 12:42 pm and 3:17 pm

Q20:

The height of a piston, β„Ž, in inches can be modeled by the equation 𝑦=2π‘₯+5cos, where π‘₯ represents the crank angle. Find the height of the piston when the crank angle is 55∘.

Q21:

The number of hours of daylight in Paris depends on the season, and it is modeled by 𝑑=12βˆ’4ο€Ό2πœ‹365(𝑑+10)cos, where 𝑑 is the number of the days in a year (January 1 is day 1). According to this model, on what days does daylight in Paris last 10 hours?

  • AFebruary 20, April 21
  • BJanuary 20, May 22
  • CAugust 21, October 21
  • DApril 21, August 21
  • EFebruary 20, October 21

Q22:

Outside temperatures over 24 hours can be modeled as a sinusoidal function. On a day with an average temperature of 85∘F, the maximum temperature of 105∘F is recorded at 5 pm. Find, to the nearest degree, the temperature at 9 am.

Q23:

A particle moves along the π‘₯-axis so that its displacement from the origin 𝑂 after 𝑑 seconds is 7(12𝑑)sin meters. Find the times at which the particle’s displacement is βˆ’72 meters. Use 𝑛 to denote an arbitrary nonnegative integer.

  • A 𝑑 = ο€» πœ‹ 7 2 + 𝑛 πœ‹ 6  s e c o n d s , 𝑑 = ο€Ό 7 πœ‹ 6 + 2 𝑛 πœ‹  s e c o n d s
  • B 𝑑 = ο€Ό 7 πœ‹ 7 2 + 𝑛 πœ‹ 6  s e c o n d s , 𝑑 = ο€Ό 7 πœ‹ 6 + 2 𝑛 πœ‹  s e c o n d s
  • C 𝑑 = ο€Ό 7 2 + 𝑛 πœ‹ 6  s e c o n d s , 𝑑 = ο€Ό 7 2 + 2 𝑛 πœ‹  s e c o n d s
  • D 𝑑 = ο€» πœ‹ 7 2 + 𝑛 πœ‹ 6  s e c o n d s , 𝑑 = ο€Ό 7 2 + 2 𝑛 πœ‹  s e c o n d s
  • E 𝑑 = ο€Ό 7 πœ‹ 7 2 + 𝑛 πœ‹ 6  s e c o n d s , 𝑑 = ο€Ό 1 1 πœ‹ 7 2 + 𝑛 πœ‹ 6  s e c o n d s

Q24:

Outside temperatures over 24 hours can be modeled as a sinusoidal function with the daily maximum occurring after midday. On a day when the average temperature is first recorded at 10 am, the temperature varies between 47∘F and 63∘F. After midnight, when is the first time the temperature reaches 51∘F?

Q25:

The sea ice area around the South Pole fluctuates between about 18 million square kilometers in September to 3 million square kilometers in March. Assuming a sinusoidal fluctuation, when are there more than 15 million square kilometers of sea ice? Give your answer as a range of dates to the nearest day.

  • Afrom July 7 to November 24
  • Bfrom January 15 to June 16
  • Cfrom July 10 to September 1
  • Dfrom July 7 to October 24
  • Efrom September 10 to February 19

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