Periodic Modeling

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Periodic Modeling · Grade 11 · Worksheet 1

1. Aroha is monitoring the height of a Ferris wheel car above the ground. The Ferris wheel has a diameter of 36 meters and its boarding platform is 3 meters above the ground. The wheel completes one full revolution every 40 seconds. At time t = 0 seconds, Aroha boards the car at the lowest point of the wheel. Write a sine function of the form h(t) = A sin(Bt + C) + D that models the height of the car above the ground (in meters) as a function of time t (in seconds). Then, determine the height of the car above the ground after 10 seconds. Answer: 21 meters Solution: Determine the amplitude A. The wheel's diameter is 36 m, so the radius is 18 m. The car oscillates 18 m above and below the center.
   Full step-by-step solution

   Step 1: Determine the amplitude A. The wheel's diameter is 36 m, so the radius is 18 m. The car oscillates 18 m above and below the center. Amplitude A = 18. Step 2: Determine the vertical shift D. The center of the wheel is at height = radius + platform height = 18 + 3 = 21 m. So D = 21. Step 3: Determine the period and angular frequency B. Period T = 40 seconds. B = 2π/T = 2π/40 = π/20. Step 4: Determine the phase shift C. At t=0, the car is at the lowest point, which is D - A = 21 - 18 = 3 m. A sine function starts at 0 and increases. To start at the minimum, we need sin(θ) = -1 at t=0. Sine equals -1 at θ = -π/2. So B*0 + C = -π/2, thus C = -π/2. Step 5: Write the function. h(t) = 18 sin(πt/20 - π/2) + 21. Step 6: Find h(10). h(10) = 18 sin(π*10/20 - π/2) + 21 = 18 sin(π/2 - π/2) + 21 = 18 sin(0) + 21 = 18*0 + 21 = 21 meters. The height of the car after 10 seconds is 21 meters.

2. Emma is tracking the height of a Ferris wheel seat. The maximum height is 75 meters, the minimum height is 15 meters, and it completes one full revolution every 18 minutes. If the seat starts at its minimum height, write the trigonometric function for the height h(t) in meters after t minutes. Answer: h(t) = 30 sin(πt/9 - π/2) + 45 Solution: Find the amplitude A. A = (max - min)/2 = (75 - 15)/2 = 60/2 = 30. Find the vertical shift D.
   Full step-by-step solution

   Step 1: Find the amplitude A. A = (max - min)/2 = (75 - 15)/2 = 60/2 = 30. Step 2: Find the vertical shift D. D = (max + min)/2 = (75 + 15)/2 = 90/2 = 45. Step 3: Find the coefficient B from the period. Period = 18 minutes. For a sine function, period = 2π/B, so 18 = 2π/B, thus B = 2π/18 = π/9. Step 4: Determine the phase shift C. The seat starts at its minimum height at t=0. A standard sine function, sin(θ), starts at 0 and increases. We need it to start at its minimum. The minimum of a sine wave occurs when its argument is -π/2 (or 3π/2, etc.). So we set Bt + C = -π/2 at t=0. This gives C = -π/2. Step 5: Write the final function. h(t) = A sin(Bt + C) + D = 30 sin((π/9)t - π/2) + 45. The final answer is h(t) = 30 sin(πt/9 - π/2) + 45.

3. Mere is modeling the height of a Ferris wheel car above the ground. The wheel has diameter 30 meters, the bottom of the wheel is 2 meters above the ground, and it completes one revolution every 90 seconds. If a car starts at the 3 o'clock position, write the trigonometric function h(t) for the car's height after t seconds. Answer: h(t) = 15sin(πt/45 - π/2) + 17 Solution: Determine amplitude A. The diameter is 30 m, so radius is 15 m. Amplitude A = 15.
   Full step-by-step solution

   Step 1: Determine amplitude A. The diameter is 30 m, so radius is 15 m. Amplitude A = 15. Step 2: Determine vertical shift D. Minimum height is 2 m, maximum height is 2 + 30 = 32 m. Midline is (2 + 32)/2 = 17 m. So D = 17. Step 3: Determine B from period. Period is 90 seconds. B = 2π/90 = π/45. Step 4: Determine phase shift C. Starting at 3 o'clock position (rightmost point) corresponds to sine function at its midline going upward. Standard sine starts at midline going upward at phase shift -π/2. So C = -π/2. Step 5: Write the function: h(t) = 15sin((π/45)t - π/2) + 17 The answer is h(t) = 15sin(πt/45 - π/2) + 17.

4. Matiu is studying the height of a passenger car on a Ferris wheel. The Ferris wheel has a diameter of 32 meters and its boarding platform is 4 meters above the ground. The wheel completes one full revolution every 60 seconds. At time t = 0 seconds, Matiu boards the car at the lowest point of the wheel. Write a sine function of the form h(t) = A sin(Bt + C) + D that models the height of the car above the ground (in meters) as a function of time t (in seconds). Then, determine the height of the car above the ground after 15 seconds. Answer: 20 meters Solution: Determine the amplitude A. The wheel's diameter is 32 m, so the radius is 16 m. The car oscillates 16 m above and below the center.
   Full step-by-step solution

   Step 1: Determine the amplitude A. The wheel's diameter is 32 m, so the radius is 16 m. The car oscillates 16 m above and below the center. Amplitude A = 16. Step 2: Determine the vertical shift D. The center of the wheel is at height = radius + platform height = 16 + 4 = 20 m. So D = 20. Step 3: Determine the period and angular frequency B. Period T = 60 seconds. B = 2π/T = 2π/60 = π/30. Step 4: Determine the phase shift C. At t=0, the car is at the lowest point, which is D - A = 20 - 16 = 4 m. A sine function starts at 0 and increases. To start at the minimum, we need sin(θ) = -1 at t=0. Sine equals -1 at θ = -π/2. So B*0 + C = -π/2, thus C = -π/2. Step 5: Write the function. h(t) = 16 sin(πt/30 - π/2) + 20. Step 6: Find h(15). h(15) = 16 sin(π*15/30 - π/2) + 20 = 16 sin(π/2 - π/2) + 20 = 16 sin(0) + 20 = 16*0 + 20 = 20 meters. The height of the car after 15 seconds is 20 meters.

5. Emma is modeling the height of a Ferris wheel seat above the ground. The maximum height is 85 meters, the minimum height is 15 meters, and it completes one full revolution every 20 minutes. If the seat starts at its minimum height at t=0, write the trigonometric function h(t) = A sin(Bt + C) + D that models the height in meters after t minutes. Answer: h(t) = 35 sin(πt/10 - π/2) + 50 Solution: Find amplitude A. A = (max - min)/2 = (85 - 15)/2 = 70/2 = 35. Find vertical shift D.
   Full step-by-step solution

   Step 1: Find amplitude A. A = (max - min)/2 = (85 - 15)/2 = 70/2 = 35. Step 2: Find vertical shift D. D = (max + min)/2 = (85 + 15)/2 = 100/2 = 50. Step 3: Find B from period. Period = 20 minutes, so 20 = 2π/B, thus B = 2π/20 = π/10. Step 4: Find phase shift C. At t=0, the seat is at minimum height. For a sine function, the minimum occurs at -π/2 (since sin(-π/2) = -1). So we need B*0 + C = -π/2, giving C = -π/2. Step 5: Write the function: h(t) = 35 sin(πt/10 - π/2) + 50. The answer is h(t) = 35 sin(πt/10 - π/2) + 50.

6. Ava is tracking the height of a buoy in the ocean. The buoy oscillates between a maximum height of 16 feet and a minimum height of 6 feet above a reference point. The buoy completes one full cycle every 11 hours. If at time t=0, the buoy is at its average height and moving upward, write the trigonometric function h(t) that models the buoy's height. Answer: h(t) = 5 sin(2πt/11) + 11 Solution: Find the amplitude (A). Amplitude = (maximum - minimum)/2 = (16 - 6)/2 = 10/2 = 5. Find the vertical shift (D).
   Full step-by-step solution

   Step 1: Find the amplitude (A). Amplitude = (maximum - minimum)/2 = (16 - 6)/2 = 10/2 = 5. Step 2: Find the vertical shift (D). D = (maximum + minimum)/2 = (16 + 6)/2 = 22/2 = 11. Step 3: Find the coefficient B using the period. Period = 11 hours = 2π/B, so B = 2π/11. Step 4: Determine the phase shift (C). Since the buoy starts at average height (11 feet) and is moving upward at t=0, we use a sine function with no horizontal shift: h(t) = A sin(Bt) + D. Step 5: Write the final function: h(t) = 5 sin(2πt/11) + 11.