Answer: The elapsed time t equals 33.0 seconds.
Detailed explanation:
We start with the equation modeling the vertical distance H between the dock and the boat's mast at time t seconds after the first peak:
H(t) = 5cos( 2π/3 t) − 35.5H
The maximum height is 5 units.
When the mast is at its lowest point, H(t) equals 0.
5cos( 2π/3 t) − (35.5/100)H = 0
5cos( 2π/3 t) − 0.355 × 5 = 0
5cos( 2π/3 t) − 0.1775 = 0
Thus, 5cos( 2π/3 t) = 0.1775
From there, we find: cos( 2π/3 t) = 0.1775/5
Which simplifies to cos( 2π/3 t) = 0.355
To solve for t, we compute: 2π/3 t = cos^{-1}(0.355)
This yields: 2π/3 t = 69.2
Therefore, by multiplying both sides by 3, we have: 2πt = 69.2 × 3
Which results in 2πt = 207.6
Finally, solving for t gives: t = 207.6/2π
This results in t = 33.0 seconds.
