The power output of a tuba is 0.39 W.
At what distance is the sound intensity of
the tuba 1.4 × 10−3 W/m2?
Answer in units of m
Really not a good question, as a tube is NOT an isotropic sound source. Ignoring that bit of reality, assumeing it is an isotropic source, then
.49=1.4E-3*4PI*distance^2
To find the distance at which the sound intensity of the tuba is 1.4 × 10^(-3) W/m^2, we can use the inverse square law equation:
I1 / I2 = (r2 / r1)^2
Where:
I1 = initial intensity (0.39 W)
I2 = final intensity (1.4 × 10^(-3) W/m^2)
r1 = initial distance (unknown)
r2 = final distance (to be calculated)
We can rearrange the equation to solve for r2:
(r2 / r1)^2 = I2 / I1
Taking the square root of both sides:
r2 / r1 = √(I2 / I1)
Now we can substitute the given values into the equation:
r2 / r1 = √((1.4 × 10^(-3) W/m^2) / (0.39 W))
Simplifying:
r2 / r1 = √(3.58974358974359 × 10^(-3))
r2 / r1 = 0.0599820022
To solve for r2, we need to isolate it:
r2 = r1 * 0.0599820022
Since r1 is the unknown distance, we don't know its actual value. Therefore, we cannot determine the exact distance at which the sound intensity of the tuba is 1.4 × 10^(-3) W/m^2 without additional information.
To find the distance at which the sound intensity of the tuba is 1.4 × 10^−3 W/m^2, we can use the inverse square law of sound. According to the inverse square law, the sound intensity is inversely proportional to the square of the distance from the sound source.
The formula for the inverse square law is:
I₁/I₂ = (r₂/r₁)²
Where:
I₁ is the initial sound intensity (0.39 W)
I₂ is the desired sound intensity (1.4 × 10^−3 W/m²)
r₁ is the initial distance from the sound source (we need to find this)
r₂ is the desired distance from the sound source (unknown)
Now let's solve the equation for r₂:
I₁/I₂ = (r₂/r₁)²
Substituting the given values, we get:
0.39 / (1.4 × 10^−3) = (r₂/r₁)²
To isolate (r₂/r₁)², we can take the square root of both sides of the equation:
√(0.39 / (1.4 × 10^−3)) = r₂/r₁
Simplifying the expression on the left side of the equation:
√(0.39 / (1.4 × 10^−3)) ≈ 142.091
Now, we can rearrange the equation to isolate r₂:
r₂ = (r₁ * √(0.39 / (1.4 × 10^−3)))
Since r₂ is the desired distance where the sound intensity is 1.4 × 10^−3 W/m², we can assume that r₁ = 1 meter.
Plugging in the values, we get:
r₂ = (1 * √(0.39 / (1.4 × 10^−3))) ≈ 10.1527
Therefore, the distance at which the sound intensity of the tuba is 1.4 × 10^−3 W/m² is approximately 10.15 meters.