Problem 17.68 8of9 You are standing 2 .5 m directly in front of one of the two l

Question

Problem 17.68 8of9 You are standing 2 .5 m directly in front of one of the two loudspeakers shown in the figure. They are 3.0 m apart and both are playing a 606 Ha tone in phase (F igure1 Part A As you begin to walk directly away lrom the speaker, at what distances from the speaker do you hear a minimuts sound intensly? The reom temperature es 20°C Express y answer numerically using two signkant ligures, " there is more than one inswer, enter your answers in ascendng ceder separated by commas Figure 1 of1 Wak Provide Feedback 2.5 m 3.0 m

Explanation / Answer

Part A -

Here, the destructive interference occurs when the difference in distance from the in phase speakers is

d = /2*(2n -1) where n = 1,2,3...

Now given that the room temperature is 20 deg C

then v = 343m/s

so = 343/686 = 0.500m

So the difference in distances d for minimum sound is -

0.500 / 2*(2n -1) = 0.250, 0.750, 1.25, 1.75, 2.25, 2.75, 3.25, 3.75, 4.25,4.75....

Therefore, we must find the distance such that the hypotenuse of the triangle minus the side = 0.250, 0.750, 1.25, 1.75, 2.25, 2.75, ,....

so sqrt(3^2 + (2.5+x)^2) - (2.5+x) = 0.250, 0.750, 1.25, 1.75, 2.25, 2.75, .....

Therefore 9 + 6.25 + 5x + x^2 = (0.25 + 2.5 + x)^2 = (2.75 + x)^2 = 7.5625 + 5.5x + x^2

So 0.5x = 15.25 - 7.5625 = 7.6875

=> x1 = 7.6875/0.5 = 15.375 m

again repeating for d = 0.750

so, 9 + 6.25 + 5x + x^2 = (0.75 + 2.5 + x)^2 = (3.25 + x)^2 = 10.5625 + 6.5x + x^2

=> 1.5x = 15.25 - 10.5625 = 4.6875

=> x2 = 4.6875/1.5 = 3.125m

again, for d = 1.250

Therefore 9 + 6.25 + 5x + x^2 = (1.25 + 2.5 + x)^2 = (3.75 + x)^2 = 14.0625 + 7.5x + x^2

So 2.5x = 15.25 - 14.0625 = 1.1875

So x3 = 1.1875/2.5 = 0.475m

repeating for d = 1.750

Therefore 9 + 6.25 + 5x + x^2 = (1.75 + 2.5 + x)^2 = (4.25 + x)^2 = 18.0625 + 9.5x + x^2

So 1.5x = 15.25 - 18.0625 = -2.8125

This value is not feasible as you would be closer to the speaker than you started

So x = 0.475, 3.125, 15.375

therefore, the distances from the speaker is (2.5 + x) = 2.975, 5.625, 17.875

In two significant figures, your answers are -

3.0 m, 5.6 m and 18.0 m

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