Periodic Function Modeling Worksheets Grade 12

1. 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.
2. A marine biologist is studying the tidal patterns in a coastal bay. The water depth D(t) in meters follows the function D(t) = 3.5 + 2.8cos(πt/6) + 1.2sin(πt/6), where t is time in hours after midnight. A research vessel requires a minimum depth of 5 meters to safely navigate the bay. During what time intervals in the first 12 hours can the vessel safely enter?
3. A Ferris wheel with a diameter of 60 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 sin(C(t + D)), where t is time in minutes after boarding. Determine the exact values of A, B, C, and D for this model.

…and 4 more problems

1. A water wheel with a diameter of 12 meters is mounted so that its lowest point is 1 meter above the water surface. The wheel rotates counterclockwise at a constant rate, completing one full revolution every 20 seconds. A bucket attached to the rim of the wheel is initially at its highest point. The height h (in meters) of the bucket above the water surface can be modeled by a sinusoidal function of the form h(t) = A + B cos(C(t - D)), where t is time in seconds after observation begins. Determine the exact values of A, B, C, and D for this situation.
2. A biologist is modeling the population of a rare bird species in a nature reserve. The population P(t) follows the function P(t) = 500e^(0.03t) / (1 + 0.2e^(0.03t)), where t is time in years since monitoring began. Determine the maximum sustainable population that the reserve can support according to this model.
3. A marine biologist is studying the vertical motion of ocean buoys during a storm. The buoy's height above sea level follows the function h(t) = 2cos(πt/4) + 3sin(πt/4), where h is in meters and t is time in seconds. Determine the exact time during the first 8 seconds when the buoy reaches its maximum height above sea level.

…and 3 more problems

1. Isabella's biorhythm alertness level varies periodically with a maximum of 92% at 2pm and minimum of 68% at 2am. Model with a cosine function A cos(B(t - C)) + D. Find A, B, C, and D.
2. Charlotte is an engineer designing a suspension bridge. The vertical displacement of a point on the main cable, measured in meters from its equilibrium position, is modeled by the periodic function y(t) = 12 sin(πt/8) + 5 cos(πt/8), where t is time in seconds. During the first 16 seconds, determine the exact time(s) when the displacement first reaches a maximum value of 13 meters.
3. Ava is monitoring the water level in a tidal estuary for her environmental science project. The depth of water D(t) in meters at a measurement station is modeled by the periodic function D(t) = 6 + 3 cos(πt/6) + 4 sin(πt/6), where t is the time in hours after midnight. A local ferry requires a minimum depth of 8 meters to dock safely. During what time interval in the first 12 hours can the ferry first dock safely?

…and 4 more problems