Periodic Function Modeling

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Periodic Function Modeling · Grade 12 · Worksheet 2

1. Aroha is an engineer monitoring a suspension bridge. The vertical displacement of a point on the main cable, in meters from its equilibrium position, is modeled by the periodic function y(t) = 14 sin(πt/9) + 9 cos(πt/9), where t is time in seconds. During the first 18 seconds, determine the exact time when the displacement first reaches a maximum value of √277 meters. Answer: t = 81/(2π) - (9/π) arctan(14/9) seconds Solution: Rewrite y(t) = 14 sin(πt/9) + 9 cos(πt/9) as R sin(πt/9 + φ). Use the identity: a sin θ + b cos θ = R sin(θ + φ), where R = sqrt(a² + b²) and φ satisfies sin φ = b/R and cos φ = a/R. Here a = 14, b = 9.
   Full step-by-step solution

   Step 1: Rewrite y(t) = 14 sin(πt/9) + 9 cos(πt/9) as R sin(πt/9 + φ). Step 2: Use the identity: a sin θ + b cos θ = R sin(θ + φ), where R = sqrt(a² + b²) and φ satisfies sin φ = b/R and cos φ = a/R. Step 3: Here a = 14, b = 9. So R = sqrt(14² + 9²) = sqrt(196 + 81) = sqrt(277). Step 4: Then sin φ = b/R = 9/√277, cos φ = a/R = 14/√277. Thus φ = arctan(9/14). Step 5: So y(t) = √277 sin(πt/9 + φ) with φ = arctan(9/14). Step 6: The maximum displacement is √277 meters, which occurs when sin(πt/9 + φ) = 1. Step 7: So πt/9 + φ = π/2 + 2πk for integer k. For the first maximum in the first 18 seconds, take k = 0. Step 8: Thus πt/9 = π/2 - φ = π/2 - arctan(9/14). Step 9: Multiply both sides by 9/π: t = (9/π)(π/2 - arctan(9/14)) = 9/2 - (9/π) arctan(9/14). Step 10: Simplify: t = 81/(18) - (9/π) arctan(9/14) = 4.5 - (9/π) arctan(9/14). Step 11: The exact time when displacement first reaches √277 meters is t = 9/2 - (9/π) arctan(9/14) seconds.

2. A Ferris wheel with a diameter of 50 meters completes one full revolution every 3 minutes. The boarding platform is 3 meters above ground level, and a passenger boards at the lowest point. The height of a passenger above ground can be modeled by a sinusoidal function h(t) = A + B cos(C(t + D)), where t is time in minutes after boarding. Determine the exact values of A, B, C, and D for this model. Answer: A=28,B=-25,C=2π/3,D=0 Solution: Determine the amplitude B. The wheel's diameter is 50 meters, so the radius is 25 meters. The amplitude is equal to the radius, so B = 25.
   Full step-by-step solution

   Step 1: Determine the amplitude B. The wheel's diameter is 50 meters, so the radius is 25 meters. The amplitude is equal to the radius, so B = 25. Since the passenger boards at the lowest point and we're using cosine, which starts at its maximum, we need a negative sign: B = -25. Step 2: Determine the vertical shift A. The center of the Ferris wheel is at height = platform height + radius = 3 + 25 = 28 meters above ground. So A = 28. Step 3: Determine the angular frequency C. The period is 3 minutes, and the period formula is T = 2π/C. So 3 = 2π/C, which gives C = 2π/3. Step 4: Determine the phase shift D. The passenger boards at the lowest point. For h(t) = 28 - 25 cos(C(t + D)), at t = 0, h(0) should be at the minimum height of 3 meters. So 28 - 25 cos(C(0 + D)) = 3. This gives 28 - 25 cos(CD) = 3, so 25 cos(CD) = 25, so cos(CD) = 1. The simplest solution is when CD = 0, so D = 0. Final answer: A = 28, B = -25, C = 2π/3, D = 0

3. Tane's ocean tide depth varies periodically with a maximum depth of 14 meters at 3:00 AM and a minimum depth of 8 meters at 9:00 AM. Model the depth d(t) as a cosine function of time t in hours since midnight, d(t) = A cos(B(t - C)) + D. Determine A, B, C, and D. Answer: A=3, B=π/6, C=3, D=11 Solution: Find the vertical shift D, which is the average of maximum and minimum depths. D = (14 + 8) / 2 = 22 / 2 = 11 Find the amplitude A, which is half the difference between maximum and minimum depths.
   Full step-by-step solution

   Step 1: Find the vertical shift D, which is the average of maximum and minimum depths. D = (14 + 8) / 2 = 22 / 2 = 11 Step 2: Find the amplitude A, which is half the difference between maximum and minimum depths. A = (14 - 8) / 2 = 6 / 2 = 3 Step 3: Determine the period. The tide completes one full cycle from max to min to max. From 3:00 AM (max) to 9:00 AM (min) is 6 hours, which is half a period. So the full period is 12 hours. Step 4: Find B using the period formula: Period = 2π/B 12 = 2π/B B = 2π/12 = π/6 Step 5: Find the horizontal shift C. For a cosine function, the maximum occurs when the argument is 0. The maximum occurs at t = 3 hours (3:00 AM), so: B(t - C) = 0 when t = 3 (π/6)(3 - C) = 0 3 - C = 0 C = 3 Step 6: Write the final function with all parameters: d(t) = 3 cos((π/6)(t - 3)) + 11 The parameters are: A = 3, B = π/6, C = 3, D = 11

4. Liam is a physicist monitoring the voltage output of an experimental alternating current generator. The voltage V(t) in volts is modeled by the periodic function V(t) = 21 sin(πt/9) + 28 cos(πt/9), where t is time in seconds. Determine the exact time during the first 18 seconds when the voltage first reaches its maximum value. Answer: t = 9/2 - (9/π) arctan(3/4) seconds, approximately 2.31 seconds Solution: We have V(t) = 21 sin(πt/9) + 28 cos(πt/9). Rewrite in the form R sin(πt/9 + φ). Use the identity: a sin θ + b cos θ = R sin(θ + φ) where R = sqrt(a^2 + b^2) and φ satisfies sin φ = b/R and cos φ = a/R.
   Full step-by-step solution

   Step 1: We have V(t) = 21 sin(πt/9) + 28 cos(πt/9). Rewrite in the form R sin(πt/9 + φ). Step 2: Use the identity: a sin θ + b cos θ = R sin(θ + φ) where R = sqrt(a^2 + b^2) and φ satisfies sin φ = b/R and cos φ = a/R. Step 3: Here a = 21, b = 28. So R = sqrt(21^2 + 28^2) = sqrt(441 + 784) = sqrt(1225) = 35. Step 4: Then sin φ = b/R = 28/35 = 4/5, and cos φ = a/R = 21/35 = 3/5. Thus tan φ = sin φ / cos φ = (4/5) / (3/5) = 4/3, so φ = arctan(4/3). Step 5: So V(t) = 35 sin(πt/9 + φ) with φ = arctan(4/3). Step 6: The maximum voltage is 35 volts, which occurs when sin(πt/9 + φ) = 1. Step 7: Set πt/9 + φ = π/2 + 2πk. For the first maximum in the first 18 seconds, take k = 0. Step 8: Then πt/9 = π/2 - φ = π/2 - arctan(4/3). Step 9: Multiply both sides by 9/π: t = (9/π)(π/2 - arctan(4/3)) = 9/2 - (9/π) arctan(4/3). Step 10: Compute numerically: arctan(4/3) ≈ arctan(1.3333) ≈ 0.9273 radians. Step 11: Then t = 4.5 - (9/π)(0.9273) ≈ 4.5 - (9/3.1416)(0.9273) ≈ 4.5 - (2.8648)(0.9273) ≈ 4.5 - 2.656 ≈ 1.844 seconds. Step 12: Verify: at t ≈ 1.844, V = 35 sin(π(1.844)/9 + 0.9273) = 35 sin(0.6435 + 0.9273) = 35 sin(1.5708) ≈ 35(1) = 35 volts. The exact time when the voltage first reaches its maximum is t = 9/2 - (9/π) arctan(4/3) seconds, approximately 1.84 seconds.

5. Aisha is analyzing the motion of a Ferris wheel for her physics project. The height of a passenger above the ground (in meters) is modeled by the function h(t) = 15sin(π/10 * t - π/2) + 18, where t is time in seconds after the ride starts. Determine the exact time(s) during the first complete revolution when the passenger reaches a height of 25 meters. Answer: t = 20/3 seconds and t = 40/3 seconds Solution: Trigonometric equations model periodic phenomena like circular motion, sound waves, and seasonal patterns.
   Full step-by-step solution

   Trigonometric equations model periodic phenomena like circular motion, sound waves, and seasonal patterns. To solve them, we isolate the trigonometric expression and use inverse functions, but must consider that these functions are periodic and typically have multiple solutions within a given interval. The general approach involves setting up the equation, using algebraic manipulation to isolate the trig function, applying inverse trig operations, and then finding all solutions within the specified domain by considering the function's periodicity and any phase shifts.

6. A large grandfather clock in a tower has a pendulum that swings back and forth. The horizontal distance (in centimeters) of the pendulum bob from the centerline of the clock is modeled by a sinusoidal function of the form d(t) = A cos(B(t - C)), where t is time in seconds after the bob is released from its maximum displacement. The pendulum completes one full swing (back and forth) every 2 seconds. The maximum horizontal distance from the centerline is 12 centimeters, and at t = 0, the bob is at its maximum displacement to the right. Determine the exact values of A, B, and C for this model. Answer: A=12, B=π, C=0 Solution: Determine the amplitude A. The amplitude is the maximum displacement from the centerline, which is given as 12 cm. So A = 12.
   Full step-by-step solution

   Step 1: Determine the amplitude A. The amplitude is the maximum displacement from the centerline, which is given as 12 cm. So A = 12. Step 2: Determine the coefficient B. The period of the pendulum is 2 seconds. For a cosine function, the period T = 2π/B. So 2 = 2π/B, which gives B = 2π/2 = π. Step 3: Determine the phase shift C. The bob is at its maximum displacement at t=0. For a cosine function of the form d(t) = A cos(B(t - C)), the maximum occurs when cos(B(t - C)) = 1, which happens when B(t - C) = 0, or t = C. Since the maximum occurs at t=0, we have 0 = C, so C = 0. Step 4: The model is d(t) = 12 cos(πt). The parameters are A = 12, B = π, C = 0.