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Worksheet: Modelling with Trigonometric Functions

Q1:

Isabella is jumping on a trampoline. Her height above the trampoline , in meters, is given by , at 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
  • B
  • C
  • D
  • E

Q2:

The outside temperature (in degrees Celsius) on a certain day was modeled with , where is the time after midnight in hours. At what times of the day was the temperature ? 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 on a point just above Earth’s atmosphere days after the summer solstice is given by . How many times per year is the daily solar irradiation 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 , where 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 minutes after they got into it is given by . 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 , where is the time of the day expressed in hours after midnight. At what times of the day was the temperature ?

  • 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:

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, , of the pod above the Thames minutes after they got into it.

  • A
  • B
  • C
  • D
  • E

Q8:

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

Q9:

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 each second. Find a function that models , the displacement of the end of the spring from its equilibrium position, in terms of , the time in seconds, after it was released.

  • A
  • B
  • C
  • D
  • E

Q10:

The height of a piston, , in inches can be modeled by the equation , where represents the crank angle. Find the height of the piston when the crank angle is .

Q11:

A spring is fixed at one end and hangs vertically. Its lower end is pulled 19 cm down from its equilibrium position and released. It performs 13 oscillations every second and, after 4 seconds, the amplitude decreases to 14 cm. Find a function that models , the displacement of the end of the spring from its equilibrium position, in terms of , the time in seconds after it is released.

  • A
  • B
  • C
  • D
  • E

Q12:

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

Q13:

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.