Periodic Modeling

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Answer Key & Explanations

Periodic Modeling · Grade 11 · Worksheet 3

1. sin²(π/6) + cos²(π/6) = ? Answer: 1 Solution: Recall the Pythagorean trigonometric identity. sin²(θ) + cos²(θ) = 1. Identify the given angle.
   Full step-by-step solution

   Step 1: Recall the Pythagorean trigonometric identity. The identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. Step 2: Identify the given angle. The angle in the problem is π/6. Step 3: Apply the identity. This means sin²(π/6) + cos²(π/6) = 1. Step 4: Conclusion. Since the identity holds for all angles, the expression equals 1. Final Answer: 1

2. Ava is modeling the height of ocean waves. The waves oscillate between a maximum height of 16 feet and a minimum height of 6 feet, with a period of 11 seconds. If she models the height h (in feet) as a function of time t (in seconds) using a sine function starting at the average height and increasing, what is the function h(t)? Answer: h(t) = 5 sin(2π/11 t) + 11 Solution: Find the amplitude A. A = (max - min)/2 = (16 - 6)/2 = 10/2 = 5. Find the vertical shift D.
   Full step-by-step solution

   Step 1: Find the amplitude A. A = (max - min)/2 = (16 - 6)/2 = 10/2 = 5. Step 2: Find the vertical shift D. D = (max + min)/2 = (16 + 6)/2 = 22/2 = 11. Step 3: Find the coefficient B using the period. The period is 11 seconds, and period = 2π/B. So, 11 = 2π/B, which means B = 2π/11. Step 4: Since the function starts at the average height and is increasing, there is no horizontal phase shift (C=0). Step 5: Write the function: h(t) = 5 sin((2π/11)t) + 11. The final answer is h(t) = 5 sin(2π/11 t) + 11.

3. Ava is modeling the height of a Ferris wheel seat above ground. The maximum height is 61 meters, minimum height is 16 meters, and it completes one revolution every 6 minutes. At t=0, the seat is at its minimum height. Write the function h(t) = A sin(Bt + C) + D. Answer: h(t) = 22.5 sin(π/3 t - π/2) + 38.5 Solution: Calculate amplitude A = (max - min)/2 = (61 - 16)/2 = 45/2 = 22.5 Calculate vertical shift D = (max + min)/2 = (61 + 16)/2 = 77/2 = 38.5 Calculate B using period = 6 minutes: B = 2π/6 = π/3 Determine phase shift C: At t=0, height is minimum, so sin(B×0 + C) = sin(C) = -1.
   Full step-by-step solution

   Step 1: Calculate amplitude A = (max - min)/2 = (61 - 16)/2 = 45/2 = 22.5 Step 2: Calculate vertical shift D = (max + min)/2 = (61 + 16)/2 = 77/2 = 38.5 Step 3: Calculate B using period = 6 minutes: B = 2π/6 = π/3 Step 4: Determine phase shift C: At t=0, height is minimum, so sin(B×0 + C) = sin(C) = -1. Therefore C = -π/2 Step 5: Write the complete function: h(t) = 22.5 sin(π/3 t - π/2) + 38.5

4. A Ferris wheel with a diameter of 40 meters completes one full revolution every 2 minutes. The boarding platform is 2 meters above ground level, and passengers board at the lowest point. If Liam boards the Ferris wheel at time t=0, write a trigonometric function h(t) that models his height above ground in meters as a function of time in minutes. Answer: h(t) = 20sin(πt - π/2) + 22 or h(t) = -20cos(πt) + 22 Solution: Modeling periodic motion with trigonometric functions involves determining amplitude from the range of motion, period from the cycle time, phase shifts from starting position, and vertical translations from equilibrium positions. The general form is h(t) = Asin(B(t - C)) + D or h(t) = Acos(B(t -…
   Full step-by-step solution

   Modeling periodic motion with trigonometric functions involves determining amplitude from the range of motion, period from the cycle time, phase shifts from starting position, and vertical translations from equilibrium positions. The general form is h(t) = Asin(B(t - C)) + D or h(t) = Acos(B(t - C)) + D, where A is amplitude, B relates to period, C is horizontal shift, and D is vertical shift.

5. Olivia is modeling the height of a swing above the ground. The swing reaches a maximum height of 15 feet and a minimum height of 5 feet. It completes one full cycle every 10 seconds. If she starts timing when the swing is at its average height and moving upward, write the sine function that models the height h (in feet) after t seconds. Answer: h(t) = 5 sin(π/5 t) + 10 Solution: Find the amplitude A. A = (max - min)/2 = (15 - 5)/2 = 10/2 = 5. Find the vertical shift D.
   Full step-by-step solution

   Step 1: Find the amplitude A. A = (max - min)/2 = (15 - 5)/2 = 10/2 = 5. Step 2: Find the vertical shift D. D = (max + min)/2 = (15 + 5)/2 = 20/2 = 10. Step 3: Find the coefficient B from the period. Period = 10 seconds. Period = 2π/B, so 10 = 2π/B, thus B = 2π/10 = π/5. Step 4: Determine the phase shift C. The general form is h(t) = A sin(Bt + C) + D. Starting at average height (10 ft) and moving upward means at t=0, h=10 and the derivative is positive. For a sine function, sin(0) = 0 gives the average value. So we want sin(C) = 0. To have an upward movement at t=0, the sine function must be increasing. The sine function increases after a zero if the phase is 0 (or 2π, etc.). So we can set C = 0. Step 5: Write the function: h(t) = 5 sin((π/5)t + 0) + 10 = 5 sin(π/5 t) + 10.

6. Emma is modeling the height of a Ferris wheel seat above the ground. The maximum height is 55 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) = 20 sin(πt/10 - π/2) + 35 Solution: Find amplitude A. A = (max - min)/2 = (55 - 15)/2 = 40/2 = 20. Find vertical shift D.
   Full step-by-step solution

   Step 1: Find amplitude A. A = (max - min)/2 = (55 - 15)/2 = 40/2 = 20. Step 2: Find vertical shift D. D = (max + min)/2 = (55 + 15)/2 = 70/2 = 35. Step 3: Find B from period. Period = 20 minutes. B = 2π/period = 2π/20 = π/10. Step 4: Find phase shift C. At t=0, height is minimum (15). For sine, minimum occurs at -π/2. So we need sin(B*0 + C) = -1, which means C = -π/2. Step 5: Write the function: h(t) = 20 sin(πt/10 - π/2) + 35. The answer is h(t) = 20 sin(πt/10 - π/2) + 35.

7. Aroha's height on a Ferris wheel follows a periodic pattern. The wheel has a diameter of 30 meters, the center is 20 meters above ground, and it completes one revolution every 3 minutes. Model her height h (in meters) as a function of time t (in minutes) using a sine function: h(t) = A sin(Bt + C) + D. Find A, B, C, and D. Answer: 15, 2π/3, -π/2, 20 Solution: Find amplitude A. The diameter is 30 m, so radius is 15 m. Amplitude A = 15.
   Full step-by-step solution

   Step 1: Find amplitude A. The diameter is 30 m, so radius is 15 m. Amplitude A = 15. Step 2: Find vertical shift D. The center is 20 m above ground, so D = 20. Step 3: Find B from period. Period = 3 minutes. B = 2π/period = 2π/3. Step 4: Find phase shift C. Using sine function, at t=0, Aroha is at the bottom (20-15=5 m). Standard sine starts at middle going up, so we need -π/2 phase shift: h(t) = 15 sin((2π/3)t - π/2) + 20. Step 5: Verify: At t=0, h(0) = 15 sin(-π/2) + 20 = 15(-1) + 20 = 5 m (bottom position). The parameters are A=15, B=2π/3, C=-π/2, D=20.