A speaker at an open-air concert emits 600 of sound power, radiated equally in all directions.

a)What is the intensity of the sound 7.00 from of the speaker?
I =0.974 W/m^2
b) What sound intensity level would you experience there if you did not have any protection for your ears?
B = =120 db
c)Earplugs you can buy in the drugstore have a noise reduction rating of 23 decibels. If you are wearing those earplugs, but your friend Phil is not, how far from the speaker should Phil stand to experience the same loudness as you?

7.00 what?

600 what?

Numbers without dimensions are useless

7.00 m

600 W

To find the intensity of the sound at a distance of 7.00 m from the speaker, we can use the inverse square law for sound propagation:

Intensity (I1) / Intensity (I2) = Distance (D2)^2 / Distance (D1)^2

Where:
Intensity (I1) = initial intensity at a known distance
Intensity (I2) = intensity at the desired distance
Distance (D1) = initial distance
Distance (D2) = desired distance

Given:
Initial intensity (I1) = 600 W
Initial distance (D1) = 0 m
Distance (D2) = 7.00 m

Using the inverse square law equation:

I1 / I2 = D2^2 / D1^2
600 / I2 = (7.00)^2 / (0)^2
600 / I2 = 49

Cross-multiply:

I2 = 600 / 49
I2 ≈ 12.24 W/m^2

Therefore, the intensity of the sound at a distance of 7.00 m from the speaker is approximately 12.24 W/m^2.

To find the sound intensity level without any ear protection, we can use the following formula:

Sound Intensity Level (L) = 10 * log10(I / I0)

Where:
I = Intensity of the sound
I0 = Reference intensity (usually 10^-12 W/m^2)

Given:
Intensity (I) = 12.24 W/m^2
I0 = 10^-12 W/m^2

Substituting the values into the formula:

L = 10 * log10(12.24 / 10^-12)
L = 10 * log10(12.24) + 10 * log10(10^12)
L ≈ 120 dB

Therefore, without any ear protection, the sound intensity level experienced would be approximately 120 dB.

If earplugs with a noise reduction rating of 23 decibels are worn, the effective sound intensity level can be reduced by that amount.

Let's assume that you are wearing earplugs with a noise reduction rating of 23 decibels, and you want your friend Phil to experience the same loudness as you.

Therefore, the effective sound intensity level for Phil (L_Phil) should be reduced by 23 decibels.

Mathematically, we can express this as:

L_Phil = L - 23

Given:
L (your sound intensity level) = 120 dB

Substituting the values into the equation:

L_Phil = 120 - 23
L_Phil = 97 dB

To calculate the distance Phil should stand from the speaker to experience the same loudness as you, we need to find the corresponding intensity using the formula:

I_Phil = I0 * 10^(L_Phil / 10)

Where:
I_Phil = Intensity of the sound for Phil
L_Phil = Sound intensity level for Phil
I0 = Reference intensity (usually 10^-12 W/m^2)

Given:
L_Phil = 97 dB
I0 = 10^-12 W/m^2

Substituting the values into the equation:

I_Phil = 10^-12 * 10^(97 / 10)
I_Phil ≈ 0.974 W/m^2

To find the distance (D_Phil) from the speaker, we can use the inverse square law equation:

I1 / I_Phil = D_Phil^2 / D1^2

Given:
I1 = 600 W
I_Phil ≈ 0.974 W/m^2
D1 = 0 m

600 / 0.974 = D_Phil^2 / 0^2
D_Phil^2 = (600 / 0.974) * (0^2)
D_Phil ≈ sqrt((600 / 0.974) * (0^2))

Therefore, Phil should stand approximately sqrt((600 / 0.974) * (0^2)) meters away from the speaker to experience the same loudness as you while wearing earplugs with a noise reduction rating of 23 decibels.

To solve this problem, we need to use the formula for sound intensity (I):

I = P / A

Where:
I is the sound intensity,
P is the sound power, and
A is the area.

Given that the speaker emits 600 W of sound power and it is radiated equally in all directions, we can calculate the sound intensity at a distance of 7.00 m from the speaker.

a) To calculate the intensity of the sound 7.00 m from the speaker, we need to consider that the sound power is equally distributed in all directions. Therefore, the sound power will be spread over the surface area of a sphere with a radius of 7.00 m.

The surface area of a sphere is given by the formula:

A = 4πr^2,

where r is the radius of the sphere.

Substituting the values into this formula, we get:

A = 4π(7.00^2) = 615.44 m^2.

Now we can use the formula for sound intensity to calculate I:

I = P / A = 600 W / 615.44 m^2 ≈ 0.974 W/m^2.

Therefore, the sound intensity 7.00 m from the speaker is approximately 0.974 W/m^2.

b) To calculate the sound intensity level, we can use the formula:

B = 10 log(I / I0),

where B is the sound intensity level and I0 is the reference intensity (usually set to 10^-12 W/m^2).

Substituting the values into this formula, we get:

B = 10 log(0.974 W/m^2 / 10^-12 W/m^2) ≈ 120 dB.

Therefore, if you did not have any protection for your ears, you would experience a sound intensity level of approximately 120 dB at a distance of 7.00 m from the speaker.

c) To determine the distance at which Phil should stand to experience the same loudness as you while wearing earplugs with a noise reduction rating of 23 decibels, we need to consider the decrease in sound intensity due to the earplugs.

The noise reduction rating (NRR) estimates the amount of sound reduction provided by earplugs. It is given in decibels (dB).

The formula to calculate the effective sound intensity level after using earplugs is:

B_eff = B_unprotected - NRR,

where B_eff is the effective sound intensity level,
B_unprotected is the sound intensity level without earplugs, and
NRR is the noise reduction rating of the earplugs.

In this case, B_unprotected is 120 dB and NRR is 23 dB.

Therefore, the effective sound intensity level after using the earplugs will be:

B_eff = 120 dB - 23 dB = 97 dB.

To find the distance at which Phil should stand to experience the same loudness, we can revert to the inverse square law.

The inverse square law states that the sound intensity decreases with the square of the distance from the source.

Using the formula:

I1 / I2 = (r2 / r1)^2,

where I1 and I2 are the sound intensities at distances r1 and r2, respectively, we can solve for the distance r2.

Let's assume that Phil is standing at a distance r2 from the speaker.

We know that the sound intensity at your position (with earplugs) is 0.974 W/m^2, and the sound intensity at Phil's position (without earplugs) is given by:

I2 = I1 / (r2 / r1)^2,

where I1 is the sound intensity at your position with earplugs (0.974 W/m^2) and r1 is the distance from the speaker where you are.

Simplifying the equation, we have:

(r2 / r1)^2 = I1 / I2,

(r2 / 7)^2 = 0.974 W/m^2 / 10^-12 W/m^2 ≈ 0.974 x 10^12.

Taking the square root of both sides of the equation, we have:

r2 / 7 = sqrt(0.974 x 10^12),

r2 = 7 * sqrt(0.974 x 10^12).

By calculating this expression, you can find the distance (r2) at which Phil should stand to experience the same loudness as you.