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

b) Determine the maximum velocity of the wave and when it occurs.

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

Ans: I tried finding the part a) by first finding the derivative, then how do i go about simplifying it. how do i find the max velocity with cos and sin in the function?

velocity is max when acceleration is zero, so find acceleration, set it to zero, and find t.

To determine the vertical displacement of the wave when the velocity is 0.8 m/s, we need to find the time values for which the velocity is 0.8 m/s.

a) Start by finding the derivative of the equation for vertical displacement with respect to time (t):

h'(t) = -1.2sin(2t) + 0.8cos(t)

Now, set h'(t) equal to 0.8:

-1.2sin(2t) + 0.8cos(t) = 0.8

Next, we can simplify the equation by rearranging terms:

-1.2sin(2t) = 0.8 - 0.8cos(t)

Now divide both sides by -1.2:

sin(2t) = (0.8 - 0.8cos(t)) / -1.2

Use the identity sin^2(t) + cos^2(t) = 1 to rewrite the equation:

sin(2t) = (0.8 - 0.8cos(t)) / -1.2
sin(2t) = (0.8 - 0.8cos(t)) / -1.2
1 - cos^2(t) = (0.64 + 0.64cos(t) - 0.64cos^2(t)) / 1.44

Simplifying further:

1 - cos^2(t) = 0.64 + 0.64cos(t) - 0.64cos^2(t)

Combine like terms:

0.36cos^2(t) - 0.64cos(t) + 0.36 = 0

This is a quadratic equation in terms of cos(t). Solve it using any appropriate method. Once you find the values of cos(t) that satisfy the equation, you can then find the corresponding values of t.

b) To determine the maximum velocity of the wave and when it occurs, we need to find the maximum value of the derivative we found earlier, h'(t).

Start by differentiating h'(t) with respect to time (t):

h''(t) = -2.4cos(2t) - 0.8sin(t)

To find the maximum value, we set h''(t) equal to 0 and solve for t:

-2.4cos(2t) - 0.8sin(t) = 0

Simplify further:

3cos(2t) = sin(t)

We can square both sides of the equation to eliminate the trigonometric terms:

9cos^2(2t) = sin^2(t)

Using the identity sin^2(t) + cos^2(t) = 1, we can rewrite the equation:

9cos^2(2t) = 1 - cos^2(2t)

Now, solve the quadratic equation for cos(2t). Once you find the values of cos(2t) that satisfy the equation, you can then find the corresponding values of t.

c) To determine when the wave first changes from a hill to a trough, we need to find the points where the vertical displacement is at a maximum or minimum. This occurs when the derivative of the displacement function, h'(t), changes sign.

Using the expression for h'(t) we found earlier:

h'(t) = -1.2sin(2t) + 0.8cos(t)

We need to find the values of t for which h'(t) changes sign. This occurs when sin(2t) = 0 or cos(t) = 1:

1) sin(2t) = 0
This means 2t = n * π, where n is an integer. Solve for t to find the values of t at which the wave first changes from a hill to a trough.

2) cos(t) = 1
This means t = 2n * π, where n is an integer. Solve for t to find the values of t at which the wave first changes from a hill to a trough.

By finding the solutions of these equations, you can determine the time values at which the wave first changes from a hill to a trough.