Question

The pressures caused by pure tones on the eardrum are sinusoidal. The change in pressure P in pounds per square foot on a person's eardrum from a pure tone at time t in seconds can be modeled using the equation P = A sin (2?ft + ?), where f is the frequency in cycles per second, and ? is the phase angle. When P is positive, there is an increase in pressure and the eardrum is pushed inward; when P is negative, there is a decrease in pressure and the eardrum is pushed outward. A graph of a tone P(t) = .008 sin [2?(254.17)t + ?/6] is shown to the right. Complete parts (a) through (c) below. (a) Determine algebraically the values of t for which P = 0 over [0, .005]. The solution set is { }. (Round to five decimal places as needed. Use a comma to separate answers as needed.)

          The pressures caused by pure tones on the eardrum are sinusoidal. The change in pressure P in pounds per square foot on a person's eardrum from a pure tone at time t in seconds can be modeled using the equation P = A sin (2?ft + ?), where f is the frequency in cycles per second, and ? is the phase angle. When P is positive, there is an increase in pressure and the eardrum is pushed inward; when P is negative, there is a decrease in pressure and the eardrum is pushed outward. A graph of a tone P(t) = .008 sin [2?(254.17)t + ?/6] is shown to the right. Complete parts (a) through (c) below.

(a) Determine algebraically the values of t for which P = 0 over [0, .005].

The solution set is { }.
(Round to five decimal places as needed. Use a comma to separate answers as needed.)

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The pressures caused by pure tones on the eardrum are sinusoidal. The change in pressure P in pounds per square foot on a person's eardrum from a pure tone at time t in seconds can be modeled using the equation P = A sin (2?ft + ?), where f is the frequency in cycles per second, and ? is the phase angle. When P is positive, there is an increase in pressure and the eardrum is pushed inward; when P is negative, there is a decrease in pressure and the eardrum is pushed outward. A graph of a tone P(t) = .008 sin [2?(254.17)t + ?/6] is shown to the right. Complete parts (a) through (c) below.

(a) Determine algebraically the values of t for which P = 0 over [0, .005].

The solution set is .
(Round to five decimal places as needed. Use a comma to separate answers as needed.)

Added by Brian M.

Introductory and Intermediate Algebra for College Students 4th

Robert Blitzer 4th Edition

Instant Answer

Solved by Expert Eric Carlsen

04/11/2023

Step 1

05]. We are given the equation: P(t) = 0.008 \sin(2 \times 254.17t + 6) To find the values of t for which P(t) = 0, we set P(t) equal to 0 and solve for t: 0 = 0.008 \sin(2 \times 254.17t + 6) Now, we can divide both sides by 0.008: 0 = \sin(2 \times 254.17t + Show more…

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Transcript

00:01 Alright, so we've been given this trigonometric sine function, and we're told that we want to find when this guy is equal to zero on the interval between zero and 0 .005.

00:17 Alright, so we're just going to go ahead and start hacking away at this.

00:22 In order to solve this equation, we need to have the sine function part by itself, so let's divide both sides by 0 .008.

00:30 When we do that, we get 0 equals the sine of 2 pi times 254 .17 t plus pi over 6, and then we need to use the arc sine on both sides.

00:49 We need to ask ourselves, when is sine going to equal zero? when is sine going to equal zero? well, that comes from our unit circle.

00:59 On our unit circle, not the best circle, we have these key points, which is the point 10, 01, negative 10, and 0, negative 1.

01:14 And we need to remember that the cosine value is x, but the y value is sine.

01:20 So these values that we have highlighted in yellow there are the sine values.

01:24 So what angles will give us a sine value of zero? well, that's going to be zero radians and pi radians.

01:32 So we have zero or pi are going to be equal to 2 pi times 254 .17 t plus pi over 6.

01:46 So we're going to set this into two different equations.

01:49 We have zero equals all of this, and we're going to solve it.

01:59 But we also have pi equals all of that, and we're going to solve it.

02:07 So we're solving this separately now.

02:12 We have to subtract pi over 6 from both sides...

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