Periodic Modeling

lessonbunny.com

Answer Key & Explanations

Periodic Modeling · Grade 11 · Worksheet 1

1. 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 of the wheel. Liam boards the Ferris wheel at time t=0. Write a trigonometric function h(t) that models Liam's height above ground (in meters) as a function of time (in minutes). Answer: h(t) = 22 - 20cos(πt) Solution: - Diameter = 40 m → radius = 20 m. - Boarding platform is 2 m above ground.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** - Diameter = 40 m → radius = 20 m. - Boarding platform is 2 m above ground. - Lowest point of wheel is at boarding height → so center of wheel is above ground by: Lowest point = center height − radius = 2 m So center height = 2 + 20 = 22 m above ground. - Period = 2 minutes for one revolution. - At t = 0, Liam is at the lowest point (height = 2 m). --- **Step 2: General form of height function** For circular motion with center at height c, radius R, starting at the lowest point, a cosine function is convenient: h(t) = c − R cos(ωt) Why? Because cos(0) = 1, so at t=0: h(0) = c − R = 22 − 20 = 2 m (correct). --- **Step 3: Find ω (angular frequency)** Period T = 2 minutes. For cosine: cos(ωt) repeats when ωT = 2π. So ω × 2 = 2π → ω = π radians per minute. --- **Step 4: Write the function** h(t) = 22 − 20 cos(πt) --- **Step 5: Check values** At t = 0: h(0) = 22 − 20×1 = 2 m (lowest point, correct). At t = 1 min: π×1 = π radians = 180°, cos(π) = −1, so h(1) = 22 − 20×(−1) = 22 + 20 = 42 m (highest point, correct). At t = 2 min: π×2 = 2π, cos(2π) = 1, so h(2) = 22 − 20 = 2 m (back to start, correct). --- **Final answer:** h(t) = 22 − 20 cos(πt)

2. A Ferris wheel has a maximum height of 61 meters and a minimum height of 16 meters. It completes one full revolution every 21 minutes. If Sophia boards the Ferris wheel at its lowest point, write the trigonometric function for her height h (in meters) after t minutes. Answer: h = 22.5 sin(2π/21 t - π/2) + 38.5 Solution: Determine the amplitude A. A = (max - min)/2 = (61 - 16)/2 = 45/2 = 22.5 Determine the vertical shift D. D = (max + min)/2 = (61 + 16)/2 = 77/2 = 38.5 Determine the coefficient B using the period.
   Full step-by-step solution

   Step 1: Determine the amplitude A. A = (max - min)/2 = (61 - 16)/2 = 45/2 = 22.5 Step 2: Determine the vertical shift D. D = (max + min)/2 = (61 + 16)/2 = 77/2 = 38.5 Step 3: Determine the coefficient B using the period. Period = 21 minutes = 2π/B, so B = 2π/21 Step 4: Since the motion starts at the minimum point (not the midline), we need a phase shift. For a sine function starting at the minimum, we can use h = A sin(Bt + C) + D. At t=0, h=16 (minimum). So 16 = 22.5 sin(C) + 38.5 Step 5: Solve for C: 22.5 sin(C) = 16 - 38.5 = -22.5, so sin(C) = -1, which means C = -π/2 (or 3π/2) Step 6: Write the final function: h = 22.5 sin(2π/21 t - π/2) + 38.5

3. A buoy bobs up and down in the ocean as waves pass. Its vertical displacement (in meters) above its resting position can be modeled by a sinusoidal function. The buoy reaches a maximum height of 1.6 meters above its resting position and a minimum height of 1.6 meters below its resting position. It completes one full up-and-down cycle every 6 seconds. At t = 0 seconds, the buoy is at its resting position and moving upward. Write a sine function of the form y = A sin(Bx + C) + D that models the buoy's displacement y (in meters) at time t (in seconds). Then, determine the buoy's displacement after 1 second, rounded to the nearest tenth of a meter. Answer: 1.4 meters Solution: Determine the amplitude A. The maximum displacement is 1.6 m above resting position, so A = 1.6. Step 2: Determine the period T = 6 seconds.
   Full step-by-step solution

   Step 1: Determine the amplitude A. The maximum displacement is 1.6 m above resting position, so A = 1.6. Step 2: Determine the period T = 6 seconds. The angular frequency B = 2π / T = 2π / 6 = π/3. Step 3: The vertical shift D = 0 because the resting position is at y = 0. Step 4: The starting condition: at t = 0, y = 0 and moving upward. For y = A sin(Bt + C), at t = 0 we have y = A sin(C) = 0, so sin(C) = 0, meaning C = 0 or π. Since the buoy is moving upward, the derivative y' = A B cos(Bt + C) at t = 0 must be positive. With A = 1.6 and B = π/3, y'(0) = 1.6 * (π/3) * cos(C). For C = 0, cos(0) = 1, so y'(0) > 0 (upward). For C = π, cos(π) = -1, so y'(0) < 0 (downward). Thus C = 0. The function is y = 1.6 sin(π t / 3). Step 5: Find displacement at t = 1 second: y(1) = 1.6 sin(π * 1 / 3) = 1.6 sin(π/3) = 1.6 * (√3 / 2) = 0.8√3. Approximating √3 ≈ 1.732, 0.8 * 1.732 = 1.3856, rounded to the nearest tenth is 1.4 meters.

4. A city's daily temperature variation follows a sinusoidal pattern. On a particular day, the temperature reaches its maximum of 28°C at 3:00 PM and its minimum of 16°C at 3:00 AM. Environmental scientist Maria is studying how temperature affects air quality and needs to determine when the temperature will first reach 25°C after 6:00 AM. At what time (to the nearest minute) will this occur? Answer: 11:24 AM Solution: The general form is f(t) = A sin(B(t - C)) + D or f(t) = A cos(B(t - C)) + D, where A is the amplitude, the period is 2π/B, D is the vertical shift, and C is the horizontal shift.
   Full step-by-step solution

   Sinusoidal functions are excellent for modeling periodic real-world phenomena like temperature variations, tides, and seasonal changes. The general form is f(t) = A sin(B(t - C)) + D or f(t) = A cos(B(t - C)) + D, where A is the amplitude, the period is 2π/B, D is the vertical shift, and C is the horizontal shift. When solving such problems, you first determine these parameters from the given maximum and minimum values and timing information, then set up the equation and solve for the time variable.

5. A Ferris wheel with a diameter of 40 meters completes one full revolution every 2 minutes. The height of a passenger above the ground can be modeled by a trigonometric function. If the boarding platform is 5 meters above ground level and a passenger boards at the lowest point, write the function h(t) that gives the passenger's height in meters after t minutes. Answer: h(t) = 25 - 20cos(πt) Solution: Diameter = 40 m → radius = 20 m. The wheel center is 20 m above the lowest point. Boarding platform is 5 m above ground, and passenger boards at lowest point.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** Diameter = 40 m → radius = 20 m. The wheel center is 20 m above the lowest point. Boarding platform is 5 m above ground, and passenger boards at lowest point. So: Lowest point height = 5 m above ground. Highest point height = 5 m + 40 m = 45 m above ground. Center height = 5 m + 20 m = 25 m above ground. --- **Step 2: Choose the trigonometric function** We start at the lowest point at t = 0. For a cosine function: cos(0) = 1 gives maximum height, but we want minimum at t = 0. So we can use **-cos** with a phase shift or vertical reflection. Height = center height + amplitude * cos(ωt + φ), choose φ so that at t=0, height is minimum. Better: h(t) = center + amplitude * cos(ωt + φ) Amplitude = radius = 20 m. Center = 25 m. At t = 0: h(0) = 25 + 20*cos(φ) = 5 (lowest point). So 25 + 20*cos(φ) = 5 → 20*cos(φ) = -20 → cos(φ) = -1 → φ = π (or 180°). So h(t) = 25 + 20*cos(π + ωt) Using cos(π + x) = -cos(x), we get: h(t) = 25 - 20*cos(ωt). --- **Step 3: Find ω** Period T = 2 minutes for one revolution. For cosine, period = 2π/ω = 2 → ω = π radians per minute. So h(t) = 25 - 20*cos(πt). --- **Step 4: Verify** At t = 0: h(0) = 25 - 20*cos(0) = 25 - 20 = 5 m (lowest point, correct). At t = 1: h(1) = 25 - 20*cos(π) = 25 - 20*(-1) = 25 + 20 = 45 m (highest point, correct). At t = 2: h(2) = 25 - 20*cos(2π) = 25 - 20*1 = 5 m (back to start, correct). --- **Final answer:** h(t) = 25 - 20cos(πt)

6. 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.