Periodic Function Modeling Worksheets Grade 12

1. A cargo ship's navigation buoy is tethered to the ocean floor. As waves pass, the buoy bobs vertically in a periodic motion. A sensor records the buoy's height above the seabed. The minimum height recorded is 12 meters, and the maximum height is 28 meters. The buoy reaches its highest point at t = 0 seconds, and then reaches its lowest point 9 seconds later. The height h(t) in meters can be modeled by a cosine function of the form h(t) = A + B cos(C t), where t is time in seconds. Determine the exact values of A, B, and C for this model.
2. Sophia's ocean tide depth varies periodically with a maximum depth of 18 meters at 2:00 AM and a minimum depth of 8 meters at 8:00 AM. Model the depth D(t) in meters as a cosine function of time t in hours since midnight.
3. A city's population growth is modeled by the function P(t) = 80000e^(0.03t), where t is the number of years after 2020. The city's infrastructure can support a maximum population of 120,000 people. Determine the year when the city's population will first exceed its infrastructure capacity.

…and 4 more problems

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.
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.
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.

…and 3 more problems

1. Noah is an engineer monitoring the voltage output of an experimental generator. The voltage V(t) in volts is modeled by the periodic function V(t) = 18 sin(πt/10) + 24 cos(πt/10), where t is time in seconds. Determine the exact time during the first 20 seconds when the voltage first reaches its maximum value.
2. Noah is a marine engineer monitoring the vertical motion of a wave energy converter. The height of the device above the sea floor, in meters, is modeled by the function h(t) = 6 sin(πt/11) + 5 cos(πt/11), where t is time in seconds. Determine the exact time during the first 11 seconds when the height first reaches its maximum value.
3. ∫(2x³ - 4x² + 3x - 1)dx from 1 to 2 = ?

…and 4 more problems