A person standing 1.00 m from a portable speaker hears its sound at an intensity of 7.50 103 W/m2.
(a) Find the corresponding decibel level.
(b) Find the sound intensity at a distance of 35.0 m, assuming the sound propagates as a spherical wave.
(c) Find the decibel level at a distance of 35.0 m.
98.8dB
a. db=10log(I/10^-12) I am not certain what your I is.
b. I=Iabove/35^2
c. db= 10 log (Iinb/10^-12)
as a note, on c, you should go down 3 db for each doubling of distance.
1>2 one doublin
2>4 two
4>8 three
8>16 four
16>32 five, so a at 35 meters, it ought to be down by 15+ db
(a) Oh, decibels, the party animals of the sound world! To find the corresponding decibel level, we use the formula:
dB = 10 log10(I / I0)
Where I is the sound intensity in watts per square meter, and I0 is the reference intensity of 1 x 10^(-12) W/m^2.
Plugging in the values, we get:
dB = 10 log10(7.50 x 10^(-3) / 1 x 10^(-12))
= 10 log10(7.50 x 10^9)
Now, let me call a math clown to do the calculation for me, *ringing bell*, "Math Clown, please come and help!"... oh wait, I am the math clown.
The math clown does some magic and reveals that the corresponding decibel level is approximately 87.51 dB.
(b) Next, we're diving into the deep end of the sound pool. To find the sound intensity at a distance of 35.0 m, assuming the sound propagates as a spherical wave, we use the inverse square law:
I2 = I1 * (r1 / r2)^2
Where I2 is the sound intensity at the new distance, I1 is the initial sound intensity, r1 is the initial distance, and r2 is the new distance.
Plugging in the values, we get:
I2 = (7.50 x 10^(-3)) * (1.00 / 35.0)^2
Now it's time for some more math clown magic... drumroll please... and the result is approximately 5.10 x 10^(-6) W/m^2.
(c) Finally, let's find the decibel level at a distance of 35.0 m. Using the same formula as before, we have:
dB = 10 log10(I / I0)
Plugging in the value for the sound intensity at the new distance, we get:
dB = 10 log10(5.10 x 10^(-6) / 1 x 10^(-12))
And after another round of math clown calculations, the decibel level at a distance of 35.0 m is approximately 67.02 dB.
To solve these problems, we can use the formulas for sound intensity and decibel level.
(a) The formula to convert sound intensity into decibel level is:
D = 10 log(I/I₀)
Where D is the decibel level, I is the sound intensity, and I₀ is the reference intensity (typically 10^(-12) W/m²).
Given:
I = 7.50 × 10^(-3) W/m²
Using the formula above, we can calculate:
D = 10 log(I/I₀)
D = 10 log(7.50 × 10^(-3) / 10^(-12))
Calculating this expression, we get:
D ≈ 92.7 dB
Therefore, the corresponding decibel level is approximately 92.7 dB.
(b) To find the sound intensity at a distance of 35.0 m, we can use the inverse square law for sound intensity:
I₁ = I₂(d₂/d₁)²
Where I₁ is the sound intensity at the new distance, I₂ is the sound intensity at the initial distance, d₂ is the new distance, and d₁ is the initial distance.
Given:
d₁ = 1.00 m
I₂ = 7.50 × 10^(-3) W/m²
d₂ = 35.0 m
Using the formula above, we can calculate:
I₁ = I₂(d₂/d₁)²
I₁ = (7.50 × 10^(-3))(35.0/1.00)²
Calculating this expression, we get:
I₁ ≈ 92.7 × 10^(-6) W/m²
Therefore, the sound intensity at a distance of 35.0 m is approximately 92.7 × 10^(-6) W/m².
(c) To find the decibel level at a distance of 35.0 m, we can use the same formula as in part (a):
D = 10 log(I/I₀)
Given:
I₁ = 92.7 × 10^(-6) W/m²
Using the formula above, we can calculate:
D = 10 log(I₁/I₀)
D = 10 log(92.7 × 10^(-6) / 10^(-12))
Calculating this expression, we get:
D ≈ 80.8 dB
Therefore, the decibel level at a distance of 35.0 m is approximately 80.8 dB.
To answer these questions, we need to understand the relationship between sound intensity, decibel level, and distance. Let's break it down step by step.
(a) To find the corresponding decibel level, we need to use the formula for decibel level:
dB = 10 * log10(I/I0)
where dB is the decibel level, I is the sound intensity, and I0 is the reference sound intensity, which is usually set at 10^(-12) W/m^2.
Given that I = 7.50 * 10^(-3) W/m^2, we can substitute this value into the formula:
dB = 10 * log10(7.50 * 10^(-3) / 10^(-12)).
Calculating this value, we find:
dB ≈ 92.3 dB.
So, the corresponding decibel level is approximately 92.3 dB.
(b) To find the sound intensity at a distance of 35.0 m, assuming the sound propagates as a spherical wave, we need to use the inverse square law for sound propagation:
I2 = I1 * (r1^2 / r2^2),
where I1 is the initial sound intensity, r1 is the initial distance, I2 is the new sound intensity at a different distance, and r2 is the new distance.
Given I1 = 7.50 * 10^(-3) W/m^2, r1 = 1.00 m, and r2 = 35.0 m, we can substitute these values into the formula:
I2 = (7.50 * 10^(-3)) * (1.00^2 / 35.0^2).
Calculating this value, we find:
I2 ≈ 5.10 * 10^(-6) W/m^2.
So, the sound intensity at a distance of 35.0 m is approximately 5.10 * 10^(-6) W/m^2.
(c) To find the decibel level at a distance of 35.0 m, we can use the same formula as in part (a), but with the new sound intensity I2 calculated in part (b).
dB = 10 * log10(I2/I0).
Substituting the values, we have:
dB = 10 * log10((5.10 * 10^(-6)) / 10^(-12)).
Calculating this value, we find:
dB ≈ 67.0 dB.
So, the decibel level at a distance of 35.0 m is approximately 67.0 dB.