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. Everything must be determined algebraically and i do not understand how to do that without seeing the zeros on the graph
https://www.jiskha.com/questions/1796520/an-ocean-grapher-measured-a-set-of-waves-during-a-storm-and-modelled-the-vertical
To determine the vertical displacement of the wave when the velocity is 0 m/s, we need to find the values of t for which the velocity is 0. In other words, we need to find the zeros of the derivative of the displacement function h(t).
Let's start by finding the derivative of h(t):
h(t) = 0.6cos(2t) + 0.8sin(t)
Differentiating both sides with respect to t:
h'(t) = -1.2sin(2t) + 0.8cos(t)
To find the zeros of h'(t), we set it equal to 0 and solve for t:
-1.2sin(2t) + 0.8cos(t) = 0
Now, since we don't have the graph to assist us, we can try to simplify this equation using trigonometric identities.
Let's use the identity sin^2(t) + cos^2(t) = 1:
Multiply the entire equation by 1/0.8:
-1.5sin(2t) + cos(t) = 0
Rearrange the terms:
cos(t) = 1.5sin(2t)
Using the identity sin(2t) = 2sin(t)cos(t), we can rewrite the equation as:
cos(t) = 1.5(2sin(t)cos(t))
Divide both sides by cos(t):
1 = 3sin(t)
Simplify:
sin(t) = 1/3
Now, to find the values of t that satisfy this equation, we can use the inverse sine function:
t = arcsin(1/3)
Using a calculator, we find that t ≈ 0.3398 rad (in radians).
Finally, substitute this value of t back into the original displacement function h(t) to find the vertical displacement of the wave:
h(0.3398) = 0.6cos(2 * 0.3398) + 0.8sin(0.3398)
Evaluate this equation using a calculator to find the numerical value of the vertical displacement of the wave when the velocity is 0 m/s.
To determine the vertical displacement of the wave when the velocity is 0 m/s, you need to find the points where the velocity of the wave is equal to 0. In this case, the velocity is the derivative of the displacement function. Let's find it step by step:
1. Start with the given displacement function: h(t) = 0.6cos(2t) + 0.8sin(t).
2. Find the velocity function by taking the derivative of h(t) with respect to time (t):
v(t) = h'(t) = -2(0.6)sin(2t) + 0.8cos(t).
3. Set the velocity function equal to 0 and solve for t:
-2(0.6)sin(2t) + 0.8cos(t) = 0.
Now, since this equation involves both sine and cosine functions, it may not be easy to solve for t algebraically. In this case, you can use numerical methods such as graphing the equation or using a calculator to find the values of t where the velocity is 0 m/s.
Once you have the values of t where the velocity is 0, you can substitute them back into the original displacement function h(t) to find the corresponding vertical displacement of the wave.