Question

An oceanographer measured an ocean
wave during a storm. The vertical
displacement, h, of the wave,
in metres, can be modelled by
h(t) = 0.8cost + 0.5sin2t, where t is the
time in seconds.
a) Determine the vertical displacement
of the wave when the velocity is
0.8 m/s.
b) Determine the maximum velocity of
the wave and when it fi rst occurs.
c) When does the wave fi rst change
from a “hill” to a “trough”? Explain.

Answer 1

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"

Here is a picture of h(t)
http://www.wolframalpha.com/input/?i=plot+y+%3D+0.8cosx+%2B+0.5sin(2x)