a sound wave is modeled with the equation y=1/4 cos 2pi/3 theta.
a.find the period.explain
b.find the amplitude.explain
To find the period of a sound wave modeled by the equation y = (1/4)cos(2/3πθ), we need to understand the mathematical representation of a wave's period.
a. The period of a sound wave can be determined using the formula T = (2π)/ω, where T represents the period and ω is the angular frequency. In this case, we can see that the angular frequency is 2/3π.
By comparing the equation with the standard form y = A*cos(Bθ), we can observe that B = 2/3π in this case. The coefficient of θ determines the angular frequency, so we plug this value into the formula:
T = (2π) / (2/3π) = (2π) * (3/2π) = 3
Thus, the period of the sound wave is 3.
b. The amplitude of a sound wave can be determined by the coefficient of the cosine function, A. In the given equation, the coefficient is 1/4.
Therefore, the amplitude of the sound wave is 1/4. It represents the maximum value of the oscillation, showing how loud or intense the sound wave is.
To find the period of a sound wave modeled with the equation y = (1/4)cos((2π/3)θ), we need to understand the relationship between the equation and the properties of a cosine function.
a. To find the period, we can refer to the standard form of a cosine function: y = Acos(Bx). In this equation, A represents the amplitude and B determines the frequency and period.
Comparing the given equation y = (1/4)cos((2π/3)θ) to the standard form, we can see that B = (2π/3). The period of a cosine function is given by the formula:
Period = (2π) / |B|
Substituting B = (2π/3) into the formula, we get:
Period = (2π) / |(2π/3)|
To simplify this expression, we need to remember that |(2π/3)| represents the absolute value of (2π/3), which is equal to (2π/3) since the value is already positive.
Thus, the period of the sound wave is:
Period = (2π) / (2π/3)
Period = (2π) * (3/2π)
Period = 3
Therefore, the period of the sound wave is 3.
Explanation: In this case, we can determine the period by comparing the coefficient of θ in the given equation with the standard form of a cosine function. By using the formula for the period of a cosine function, we can calculate the period of the sound wave.
b. To find the amplitude of the sound wave modeled by the equation y = (1/4)cos((2π/3)θ), we can again refer to the standard form of a cosine function: y = Acos(Bx). In this equation, A represents the amplitude.
Comparing the given equation y = (1/4)cos((2π/3)θ) to the standard form, we can see that A = 1/4.
Therefore, the amplitude of the sound wave is 1/4.
Explanation: In this case, we can determine the amplitude by simply looking at the coefficient of cosθ in the given equation. The coefficient, 1/4, represents the amplitude of the sound wave.
y = A cos(kx) has
amplitude A
period 2pi/k
Now just plug in your numbers