an ocean grapher measured a set of waves during a storm and modelled the vertical displacement of waves using h(t)=0.6cos2t+0.8sint. Determine the vertical displacement of the wave when the velocity is 0m/s.
To determine the vertical displacement of the wave when the velocity is 0 m/s, we can find the derivative of the given function h(t) with respect to time (t) and then solve for t when the derivative is 0.
The given function is:
h(t) = 0.6cos(2t) + 0.8sin(t)
Taking the derivative of h(t) with respect to t, we get:
h'(t) = -1.2sin(2t) + 0.8cos(t)
Now, set h'(t) equal to 0 and solve for t:
-1.2sin(2t) + 0.8cos(t) = 0
To solve this equation, we can use the trigonometric identity:
sin(2t) = 2sin(t)cos(t)
-1.2(2sin(t)cos(t)) + 0.8cos(t) = 0
Simplifying further:
-2.4sin(t)cos(t) + 0.8cos(t) = 0
Factoring out cos(t), we get:
cos(t)(-2.4sin(t) + 0.8) = 0
Setting each factor equal to zero:
cos(t) = 0 or -2.4sin(t) + 0.8 = 0
For cos(t) = 0, t can be π/2, 3π/2, etc., since cosine is zero at these points.
For -2.4sin(t) + 0.8 = 0, we can solve for sin(t):
-2.4sin(t) = -0.8
sin(t) = 0.8 / 2.4
sin(t) = 1 / 3
Using the unit circle, we can determine the angle (t) where sin(t) = 1/3. It is approximately 0.34 radians or about 19.47 degrees.
So, we have two values for t: π/2 and approximately 0.34 radians (or about 19.47 degrees).
Now, substitute these values of t into the original function h(t) to find the vertical displacement of the wave when the velocity is 0 m/s:
For t = π/2:
h(π/2) = 0.6cos(2(π/2)) + 0.8sin(π/2)
= 0.6cos(π) + 0.8sin(π)
= 0.6(-1) + 0.8(1)
= -0.6 + 0.8
= 0.2
For t = 0.34 radians (or about 19.47 degrees):
h(0.34) ≈ 0.6cos(2(0.34)) + 0.8sin(0.34)
≈ 0.6cos(0.68) + 0.8sin(0.34)
You can use a scientific calculator or mathematical software to find the approximate value of h(0.34) by evaluating the trigonometric functions.
Therefore, the vertical displacement of the wave when the velocity is 0 m/s is approximately 0.2 and the value for t is π/2 and 0.34 radians (or about 19.47 degrees).