Posted by Farah on Saturday, November 27, 2010 at 12:01pm.
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 = cos1(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?

Math  app of sinusoial derivatives (help+check)  drwls, Saturday, November 27, 2010 at 1:37pm
a) is correct for a while, but you cannot go from
cost(2.4sint+0.8) = 0.8
to an assumption that one or the other factor is 0.8. That only works if the product is zero, in which case either factor must be zero.
Rewrite as
cost(3sint +1) = 1
One solution to that is t = 0.
Another (obtained numerically) is about at 4.97 radians.
For (c), look for the value of t where h(t) = 0. It will be changing from a hill to a trough there. You can rewite the h(t) equation as a quadratic in sin t.
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