wave is modeled by the function ...
h(t) = 3cos(p/10)*t
What is the period of the wave (T)
*frequency (F) is defined as the number of cycles of the motion per second. What is the relationship between F and T?
Find the wave height (H) which is the vertical distance between the trough and the crest of the wave.
To find the period (T) of the wave, we need to determine the time it takes for the wave to complete one full cycle. The period is given by the formula T = 2π / (angular frequency). In this case, the angular frequency is p/10.
So we can substitute the angular frequency into the formula to find the period:
T = 2π / (p/10)
To simplify this expression, we can multiply the numerator and denominator by 10/p:
T = 2π * (10/p)
Simplifying further, we get:
T = 20π / p
Now let's move on to the relationship between the frequency (F) and the period (T). Frequency is defined as the number of cycles of the motion per second. It is the reciprocal of the period. Therefore, we have the following relationship:
F = 1 / T
To find the wave height (H), we need to understand that the function h(t) = 3cos(p/10)*t represents the vertical displacement of the wave at a given time t. The wave height (H) is the vertical distance between the trough and the crest of the wave, also known as the amplitude. In this case, the amplitude is 3. Therefore, the wave height (H) is 3.
To find the period of the wave (T), we need to look at the equation h(t) = 3cos((π/10)t). The period of a wave is the length of one complete cycle of the motion. In this equation, the coefficient of t, which is (π/10), determines the speed at which the wave oscillates. The general form of a cosine function is given by:
h(t) = A*cos(Bt + C)
Where A represents the amplitude, B represents the frequency, and C represents any horizontal shift of the wave. In our case, A = 3 and B = (π/10).
For a general cosine function, the period (T) is given by the formula:
T = 2π / |B|
Applying this formula to our equation, we have:
T = 2π / |(π/10)|
To simplify this further, we can divide both the numerator and denominator by π:
T = 2 / (1/10)
Simplifying, we get:
T = 20
Therefore, the period of the wave is 20.
Now let's discuss the relationship between frequency (F) and period (T). Frequency is defined as the number of cycles of the motion per second. It represents how many complete cycles the wave undergoes in one second. The formula for frequency is:
F = 1 / T
where T is the period of the wave. In simple terms, frequency is the reciprocal of the period. So, if we know the period, we can calculate the frequency by taking the reciprocal of the period.
Now, let's find the wave height (H), which is the vertical distance between the trough and the crest of the wave. In this case, the amplitude of the wave, represented by A in our equation h(t) = 3cos((π/10)t), is 3. The amplitude of a wave determines its maximum displacement from the equilibrium position. Therefore, the wave height (H) is equal to the amplitude of the wave, which in this case is 3.
So, the wave height (H) is 3.