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) 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.

Answer 2

The first two answers are wrong because when you have an equation that is 1= (x)(y) you cannot split it into 1=x and 1=y. Simply because when you do that although you find when x=1 when x=1 that doesn't mean y=1. The second answer is wrong because they are looking for maximum velocity which is found by taking the second derivative and setting it equal to 0 and subbing that back into the first derivative.