Question

Suppose a sound wave has a period of 0.005 seconds and an amplitude of 2 pascals. Explain whether the function P(t) = 2sin(200πt) could be used to model the sound wave, where P is the air pressure in Pascals from the sound wave after t seconds.

Which of the following statements is true?

A) No, since the period is 0.005 seconds, the equation of the function is P(t) = 2sin(400πt).

B) Yes, because an amplitude of 2 means that the coefficient of sine is 2, and a period of 0.005 means that the coefficient of t must be π * 0.005 = 200π.

C) No, since the amplitude is 2, the coefficient of sine is 1. Since the period is 0.005 seconds, the equation of the function is P(t) = sin(200πt).

D) No, since the period is 0.005 seconds, the equation of the function is P(t) = 2sin(0.005πt)

Answer 1

The correct equation for a sound wave with a period of 0.005 seconds and an amplitude of 2 Pascals is P(t) = 2sin(400πt). This means the given function P(t) = 2sin(200πt) does not correctly model the wave, as the correct angular frequency must be 400π for a period of 0.005 seconds.

Step-by-step explanation:

When modeling a sinusoidal sound wave, a few key properties must be taken into account: the amplitude and the period of the wave. The amplitude is the maximum displacement from the equilibrium position (in this case, it represents the maximum pressure variation which is 2 Pascals), while the period is the time it takes for one complete cycle of the wave. The equation given for sound pressure level P(t) = 2sin(200πt) suggests an amplitude of 2, which matches the given amplitude of the wave. However, the period of the wave depends on the coefficient of t in conjunction with 200π; since the period (T) is given by the relation T = 2π/ω, where ω is the angular frequency. For a period of 0.005 seconds, we need to find the right angular frequency ω so that 0.005 = 2π/ω, this gives ω = 2π/0.005, which is ω = 400π. Therefore, for the given period of 0.005 seconds, the wave function should be P(t) = 2sin(400πt).

Thus, the correct statement from the given options would be:

A) No, since the period is 0.005 seconds, the equation of the function is P(t) = 2sin(400πt).