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Worksheet: Modeling with Trigonometric Functions

Q1:

Isabella is jumping on a trampoline. Her height β„Ž above the trampoline , in meters, is given by β„Ž = 1 βˆ’ ο€» πœ‹ 2 𝑑  c o s , at 𝑑 seconds after she started jumping.

How many seconds after each rebound does it take her to reach a height of 50 cm during the descent? Round your answer to the nearest tenth of a second.

  • A 2.6 s
  • B 0.7 s
  • C 0.3 s
  • D 3.3 s
  • E 1.7 s

What fraction of the time is Isabella at least 1.2 m above the trampoline? Express your answer as a percentage correct to one decimal place.

  • A 4 3 . 6 %
  • B 5 6 . 4 %
  • C 7 1 . 8 %
  • D 2 8 . 2 %
  • E 8 7 . 2 %

Q2:

The outside temperature (in degrees Celsius) on a certain day was modeled with 𝑇 = 1 2 + 7 ο€» πœ‹ 1 2 ( 𝑑 βˆ’ 1 0 )  s i n , where 𝑑 is the time after midnight in hours. At what times of the day was the temperature 1 0 ∘ C ? Give your answer to the nearest minute using a 24-hour format.

  • A20:54, 23:06
  • B11:06, 20:54
  • C8:54, 11:06
  • D8:54, 23:06
  • E17:06, 2:54

Q3:

The daily solar irradiation 𝑄 ο€Ή /  W m 2 on a point just above Earth’s atmosphere 𝑛 days after the summer solstice is given by 𝑄 ( 𝑛 ) = 1 3 6 0 + 4 6 ο€Ό 2 πœ‹ 3 6 5 𝑛  c o s . How many times per year is the daily solar irradiation 1 3 5 0 W/m2? How many days after the solstice does this occur?

  • Atwice, after 79 and 104 days
  • Btwice, after 79 and 287 days
  • Ctwice, after 261 and 287 days
  • Dtwice, after 104 and 261 days
  • Etwice, after 196 and 352 days

Q4:

The number of hours of daylight in Paris depends on the season, and it is modeled by 𝑑 = 1 2 βˆ’ 4 ο€Ό 2 πœ‹ 3 6 5 ( 𝑑 + 1 0 )  c o s , where 𝑑 is the number of the day in a year (January 1 is day 1). According to this model, when is the length of the day in Paris 10 hours?

  • AFebruary 20, April 21
  • BApril 21, August 21
  • CAugust 21, October 21
  • DFebruary 20, October 21
  • EJanuary 20, May 22

Q5:

Scarlett and her friends got on a Ferris wheel. When they entered the cabin at 3:15 pm, they were 5 m above the ground. The height of the cabin 𝑑 minutes after they got into it is given by β„Ž = 2 5 βˆ’ 2 0 πœ‹ 𝑑 1 0 c o s . When were they 40 m above the ground? Give your answer to the nearest minute.

  • A 3:23 pm, 3:17 pm
  • B 3:17 pm, 3:33 pm
  • C 3:27 pm, 3:33 pm
  • D 3:23 pm, 3:27 pm
  • E 3:28 pm, 3:32 pm

Q6:

The temperature fluctuation on a cold winter’s day (in degrees Celsius) is modeled by 𝑇 = 3 ο€» πœ‹ 1 2 ( 𝑑 βˆ’ 1 4 )  + 2 c o s , where 𝑑 is the time of the day expressed in hours after midnight. At what times of the day was the temperature 0 ∘ C ?

  • A 5:13 pm, 10:47 pm
  • B 5:13 pm, 10:47 am
  • C 5:13 am, 10:47 am
  • D 5:13 am, 10:47 pm
  • E 4:47 am, 11:13 am

Q7:

A Ferris wheel is 45 m in diameter. A ride takes 10 minutes and consists of one complete revolution, starting and finishing at the lowest point. When riders board the Ferris wheel, their seats are 4 m above the ground. How much of a ride is spent more than 17 m above ground? Give your answer to the nearest second.

  • A 2 minutes 53 seconds
  • B 5 minutes 30 seconds
  • C 4 minutes 53 seconds
  • D 6 minutes 23 seconds
  • E 3 minutes 14 seconds

Q8:

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 18 oscillations every second, and the amplitude of the oscillations decreases by 1 5 % each second. Find a function that models 𝐷 , the displacement of the end of the spring from its equilibrium position, in terms of 𝑑 , the time in seconds, after it was released.

  • A 𝐷 ( 𝑑 ) = 1 0 ( 0 . 1 5 ) ( 3 6 πœ‹ 𝑑 ) 𝑑 c o s
  • B 𝐷 ( 𝑑 ) = 1 0 ( 0 . 8 5 ) ( 1 6 πœ‹ 𝑑 ) 𝑑 c o s
  • C 𝐷 ( 𝑑 ) = 1 0 ( 0 . 1 5 ) ( 1 6 πœ‹ 𝑑 ) 𝑑 c o s
  • D 𝐷 ( 𝑑 ) = 1 0 ( 0 . 8 5 ) ( 3 6 πœ‹ 𝑑 ) 𝑑 c o s
  • E 𝐷 ( 𝑑 ) = 1 0 ( 1 . 1 5 ) ( 3 6 πœ‹ 𝑑 ) 𝑑 c o s

Q9:

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

Q10:

Outside temperatures over 24 hours can be modeled as a sinusoidal function. On a day with an average temperature of 7 0 ∘ F , the maximum temperature of 8 4 ∘ F is recorded at 6 pm. Find, to the nearest degree, the temperature at 7 am.

Q11:

In a certain location, the temperature over the course of a day varies between a minimum of 6 4 ∘ F at 6 am and a maximum of 8 6 ∘ F . If the temperature is modeled by a sinusoidal function, what is the first time in the day when the temperature is 7 0 ∘ F ?

  • A4:12 am
  • B7:49 am
  • C11:49 pm
  • D1:49 am
  • E10:12 am

Q12:

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?

Q13:

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.

Q14:

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 , where is the time, in hours, after any high tide.

At what time was the next high tide?

  • A2:00 am
  • B2:30 pm
  • C2:00 pm
  • D2:30 am
  • E12:00 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 12:42 pm and 3:17 pm
  • Cbetween 3:42 pm and 6:17 pm
  • Dbetween 3:17 pm and 1:12 am
  • Ebetween 1:17 pm and 11:12 pm

Q15:

Outside temperatures over 24 hours can be modeled as a sinusoidal function. On a day with an average temperature of 8 5 ∘ F , the maximum temperature of 1 0 5 ∘ F is recorded at 5 pm. Find, to the nearest degree, the temperature at 9 am.

Q16:

A particle moves along the -axis so that its displacement from the origin after seconds is meters. Find the times at which the particle’s displacement is meters. Use to denote an arbitrary non-negative integer.

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Q17:

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 4 7 ∘ F and 6 3 ∘ F . After midnight, when is the first time the temperature reaches 5 1 ∘ F ?

Q18:

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

Q19:

The slope for a wheelchair ramp for a house has to be 1 1 2 . If the vertical distance from the ground to the door bottom is 2.5 ft, find the distance the ramp has to extend from the house in order to comply with the needed slope.

Q20:

Benjamin sits on a pier, his feet dangling 60 cm below the pier. The pier is usually 80 cm above the lake. But this is a windy day, and waves make the depth of the lake oscillate. The depth of the lake under the pier is given, in meters, by 𝑑 = 𝑑 + 0 . 3 ο€Ό 2 πœ‹ 1 0 𝑑  0 s i n , where 𝑑 0 is the depth of the lake on a quiet day, and 𝑑 is the time in seconds. What fraction of the time does Benjamin have his feet under water? Express your answer as a percentage correct to one decimal place.

  • A 2 3 . 2 %
  • B 5 3 . 5 %
  • C 7 6 . 8 %
  • D 2 6 . 8 %
  • E 3 8 . 4 %

Q21:

The sea ice area around the North Pole fluctuates between about 6 million square kilometers on September 1 to 14 million square kilometers on March 1. Assuming a sinusoidal fluctuation, when are there less than 9 million square kilometers of sea ice? Give your answer as a range of dates to the nearest day.

  • Afrom June 12 to November 13
  • Bfrom January 15 to June 16
  • Cfrom January 12 to June 13
  • Dfrom June 15 to November 16
  • Efrom September 10 to February 19

Q22:

A spring is fixed at one end and hangs vertically. Its lower end is pulled 21 cm down from its equilibrium position and released. It performs 20 oscillations every second, and, after 6 seconds, the amplitude of the oscillations is 4 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.

Q23:

Olivia and Benjamin are at home. They see a bucket suspended on an elastic string going up and down in front of their window. They record the times at which they saw the bucket appear, disappear, appear again, and so on, as shown in the table.

Time (s) 0 1 3 4 10 11
Position of the Bucket top of the window bottom of the window bottom of the window top of the window top of the window bottom of the window

What is the period 𝑇 of the bucket’s movement?

The vertical displacement of the bucket with respect to the center of its movement can be modeled with the function β„Ž = β„Ž ο€Ό 2 πœ‹ 𝑇 𝑑 + 3 πœ‹ 5  0 c o s , where 𝑇 is the period above and 𝑑 , in seconds, is measured as in the table. Given that the height of Olivia and Benjamin’s window is 1.2 m, what is the amplitude β„Ž 0 of the function? Give your answer to one decimal place.

At what value of 𝑑 is the bucket at its highest point?