Residents of Hawaii are warned of the approach of a tsunami by sirens mounted on the tops of towers. Suppose a siren produces a sound that has an intensity level of 120 dB at a distance of 2.0 m. How far away can the siren be heard?
To determine how far away the siren can be heard, we need to calculate the distance at which the sound level reaches a certain threshold. In this case, let's assume a threshold of 90 dB, which is commonly used as a reference for audible sound.
Sound intensity decreases as the square of the distance from the source increases. This relationship is described by the inverse square law:
I₁/I₂ = (r₂/r₁)²
Where I₁ and I₂ are the intensities at distances r₁ and r₂, respectively.
By rearranging the equation, we can solve for the distance r₂:
r₂ = √((I₁/I₂) * r₁²)
Given that the intensity level of the siren is 120 dB at a distance of 2.0 m, and assuming the threshold for audible sound is 90 dB, we can substitute these values into the equation:
r₂ = √((10⁽⁺⁻⁹⁰⁻¹²⁰⁾) * (2.0)²)
Simplifying this expression, we have:
r₂ = √((10⁽⁺⁻³⁰⁾) * (4.0))
r₂ = √((10⁽⁻³⁰⁾) * (4.0))
r₂ = √(10⁽⁻³⁰⁺√(4.0)))
r₂ = √(10⁽⁻³⁰⁻⁴⁰⁾)
r₂ = 10^((⁻³⁰⁻⁴⁰)/20)
Using a calculator, we can find that r₂ ≈ 10⁻⁷ meters.
Therefore, the siren can be heard at a distance of approximately 0.0000001 meters, which is equivalent to 0.1 micrometers.
To determine how far away the siren can be heard, we need to use the inverse square law, which states that the intensity of sound decreases inversely proportional to the square of the distance from the source.
Given:
Intensity level at 2.0 m = 120 dB
Let's use the following formula to find the distance at which the siren can be heard:
Intensity level 1 / Intensity level 2 = (distance 2)^2 / (distance 1)^2
Let's assume that the siren can be heard at distance x. Rearranging the formula, we have:
(distance 2)^2 = (distance 1)^2 * (Intensity level 1 / Intensity level 2)
Plugging in the given values:
x^2 = (2.0 m)^2 * (120 dB / x dB)
Now, let's solve for x:
x^2 = 4.0 m^2 * (120 dB / x dB)
x^2 = 480.0 m^2 dB / x dB
x^3 = 480.0 m^2 dB
x = ∛(480.0 m^2 dB)
x ≈ 7.58 m
Therefore, the siren can be heard up to approximately 7.58 meters away.
r =2 m.
B = 120 dB = 10•log(I2/I1)
I2 =I1•10^12.
The siren can be heard until B = 0 or
10•logx = -120.
x=10^-12,
(2/r)² =10^-12
2/r = 10^-6,
r=2•10^-6 m.