Question

An oceanographer measured a set of sea waves during a storm and modelled the vertical displacement of waves in meters using the equation h(t)=0.6cos2t+0.8sint, where t is the time in seconds.

a) Determine the vertical displacement of the wave when the velocity is 0.8m/s
Ans: -1.2sin2t+0.8cost = 0.8
-2.4(sint)(cost)+0.8cost = 0.8
cost(-2.4sint+0.8) = 0.8
cost = 0.8
t = cos-1(0.8) OR -2.4sint +0.8 = 0.8
=0.6 t = 0

b) Determine the maximum velocity of the wave and when it occurs.
Ans: cost(-2.4sint+0.8)=0
therefore t= 1.5 and 0.3 and Vmax occurs at t=1.5s

c) When does the wave first change from a hill to a trough? Explain.

Please check the above answers and help if they are incorrect, and need guidance with part c, is it asking for the height?

Answer 1

a) h(t) = 0.8cost + 0.5sin2t

v(t) = -.8sint + .5cos(2t)(2) = -.8sint + cos(2t)

= .8

-.8sint + cos(2t) = .8

time -10

8sint - 10cos(2t) = -8

4sint - 5cos(2t) = -4 , but cos(2t) = 2sin^2 t - 1

4sint - 5(2sin^2 t - 1) + 8 = 0

4sint - 10sin^2 t + 5 + 8 = 0

10sin^2 t - 4sint - 13 = 0

sint = (4 ± √536)/20 = 1.35 or -.95759... , but sint ≥ 0

sint = -.95759..

t = 1.2785 + π = appr 4.42 seconds

plug that into h(t) to get the displacement

b) take the derivative of v(t), set it equal to zero and solve

c) your turn, use the properties of what happens at "hill" and "troughs"