Wei is standing in wavy water and notices the depth of the waves varies in a periodic way that can be modeled by a trigonometric function. He starts a stopwatch to time the waves. After 1.1 seconds, and then again every 3 seconds, the water just touches his knees. Between peaks, the water recedes to his ankles. Wei's ankles are 12cm off the ocean floor, and his knees are 55cm off the ocean floor. Find the formula of the trigonometric function that models the depth D of the water t seconds after Wei starts the stopwatch. Define the function using radians.

Question

Wei is standing in wavy water and notices the depth of the waves varies in a periodic way that can be modeled by a trigonometric function. He starts a stopwatch to time the waves. After 1.1 seconds, and then again every 3 seconds, the water just touches his knees. Between peaks, the water recedes to his ankles. Wei's ankles are  12cm off the ocean floor, and his knees are  55cm off the ocean floor. Find the formula of the trigonometric function that models the depth  D of the water  t seconds after Wei starts the stopwatch. Define the function using radians.

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Determine Amplitude: Determine the amplitude of the wave. The amplitude is half the distance between the maximum and minimum values of the wave. The maximum depth is at Wei's knees (55 cm) and the minimum depth is at his ankles (12 cm). Amplitude (A) = (Maximum depth - Minimum depth) / 2 A = (55 cm - 12 cm) / 2 A = 43 cm / 2 A = 21.5 cmDetermine Vertical Shift: Determine the vertical shift of the wave. The vertical shift is the average of the maximum and minimum values of the wave. Vertical shift (D0) = (Maximum depth + Minimum depth) / 2 D0 = (55 cm + 12 cm) / 2 D0 = 67 cm / 2 D0 = 33.5 cmDetermine Period: Determine the period of the wave. The period (T) is the time it takes for the wave to complete one full cycle. Wei notices the water touches his knees every 3 seconds, so the period is 3 seconds.Convert to Radians: Convert the period from seconds to radians. The period in radians is given by 2π radians for a full cycle. Since the period is 3 seconds, we need to find how many radians correspond to 3 seconds. T (in radians) = 2π / Period (in seconds) T (in radians) = 2π / 3Determine Phase Shift: Determine the phase shift of the wave. The phase shift is the horizontal shift of the wave. Since the wave touches Wei's knees at t = 1.1 seconds, and this is the first time it happens after he starts the stopwatch, the phase shift (φ) will be negative and correspond to this time. Phase shift (φ) = -1.1 secondsWrite Trig Function: Write the formula for the trigonometric function. We will use the cosine function because it starts at a maximum point, which corresponds to the water touching Wei's knees when he starts the stopwatch. The general form of the trigonometric function is: D(t) = A * cos(B(t - φ)) + D0 Where: A is the amplitude, B is the frequency (B = 2π / T), φ is the phase shift, D0 is the vertical shift. Now we substitute the values we found: D(t) = 21.5 * cos((2π / 3)(t - (-1.1))) + 33.5 D(t) = 21.5 * cos((2π / 3)(t + 1.1)) + 33.5

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