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A toy boat is bobbing on the water. Its distance D(t) (in m ) from the floor of the lake as a function of time t (in seconds) can be modeled by a sinusoidal expression of the form a*sin(b*t)+d. At t=0, when the boat is exactly in the middle of its oscillation, it is 1m above the water's floor. The boat reaches its maximum height of 1.2m after (pi)/(4) seconds. Find D(t). t should be in radians. D(t)=◻

A toy boat is bobbing on the water.\newlineIts distance D(t) D(t) (in m \mathrm{m} ) from the floor of the lake as a function of time t t (in seconds) can be modeled by a sinusoidal expression of the form a⋅sin⁡(b⋅t)+d a \cdot \sin (b \cdot t)+d .\newlineAt t=0 t=0 , when the boat is exactly in the middle of its oscillation, it is 1 m 1 \mathrm{~m} above the water's floor. The boat reaches its maximum height of 1.2 m 1.2 \mathrm{~m} after π4 \frac{\pi}{4} seconds.\newlineFind D(t) D(t) .\newlinet t should be in radians.\newlineD(t)=□ D(t)=\square

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Q. A toy boat is bobbing on the water.\newlineIts distance D(t) D(t) (in m \mathrm{m} ) from the floor of the lake as a function of time t t (in seconds) can be modeled by a sinusoidal expression of the form a⋅sin⁡(b⋅t)+d a \cdot \sin (b \cdot t)+d .\newlineAt t=0 t=0 , when the boat is exactly in the middle of its oscillation, it is 1 m 1 \mathrm{~m} above the water's floor. The boat reaches its maximum height of 1.2 m 1.2 \mathrm{~m} after π4 \frac{\pi}{4} seconds.\newlineFind D(t) D(t) .\newlinet t should be in radians.\newlineD(t)=□ D(t)=\square
  1. Identify key points: Identify the key points from the problem.\newlineThe boat is in the middle of its oscillation at t=0t=0, which means the sinusoidal function is at its vertical shift dd, and dd is given as 1 m1\,\text{m}. The maximum height of the boat is 1.2 m1.2\,\text{m}, which occurs at t=π4t=\frac{\pi}{4} seconds. Since the maximum height is 1.2 m1.2\,\text{m} and the middle of the oscillation is at 1 m1\,\text{m}, the amplitude aa is the difference between these two heights.
  2. Calculate amplitude: Calculate the amplitude aa. Amplitude a=Maximum height−Vertical shift da = \text{Maximum height} - \text{Vertical shift } d a=1.2 m−1 ma = 1.2\,\text{m} - 1\,\text{m} a=0.2 ma = 0.2\,\text{m}
  3. Determine period: Determine the period of the sinusoidal function.\newlineSince the boat reaches its maximum height at t=π4t=\frac{\pi}{4} seconds, and this corresponds to one-quarter of the sinusoidal period (because the maximum height is reached at a quarter of the period for a sine function), we can find the full period TT by multiplying π4\frac{\pi}{4} by 44.\newlineT=π4×4T = \frac{\pi}{4} \times 4\newlineT=πT = \pi
  4. Calculate value of b: Calculate the value of b, which is related to the period TT by the formula b=2πTb = \frac{2\pi}{T}.
    b=2πTb = \frac{2\pi}{T}
    b=2ππb = \frac{2\pi}{\pi}
    b=2b = 2
  5. Write sinusoidal function: Write the sinusoidal function using the values of aa, bb, and dd.
    D(t)=asin⁡(b⋅t)+dD(t) = a\sin(b\cdot t) + d
    D(t)=0.2sin⁡(2⋅t)+1D(t) = 0.2\sin(2\cdot t) + 1

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