Question

The low-frequency speaker of a stereo set produces 10.0 W of acoustical power. If the speaker projects sound uniformly in all directions, at what distance from the speaker is the intensity level 80.0 dB

Answer 1

the required distance is 89.125 m

Step-by-step explanation:

Given the data in the question;

we know that, sound intensity B in decibels of sound is;

β(dB) = 10log₁₀(
`I` /
`I_0` )

where intensity
`I` = power / area carried by wave

`I_0` = 10⁻¹² W/m² { minimum threshold intensity }

Now,

intensity
`I` = power / area carried by wave = P/A = P/4πr² { spherical }

given that; β = 80.0 dB and P = 10 W

so

β(dB) = 10log₁₀(
`I` /
`I_0` )

we substitute

80 = 10log₁₀( P / 4πr²×
`I_0`)

80 = 10log₁₀( 10 / 4πr²× 10⁻¹² )

8 = log₁₀(10) - log₁₀( 4πr²× 10⁻¹² )

8 = 1 - log₁₀( 4πr²× 10⁻¹² )

8 - 1 = -log₁₀( 4πr²× 10⁻¹² )

7 = -log₁₀( 1.2566 × 10⁻¹¹ × r² )

7 = -[ log₁₀( 1.25 × 10⁻¹¹) + log₁₀( r² ) ]

7 = -[ -10.9 + log₁₀( r² ) ]

7 = 10.9 - log₁₀( r² )

-log₁₀( r² ) = 7 - 10.9

-log₁₀( r² ) = - 3.9

log₁₀( r² ) = 3.9

2log₁₀r = 3.9

log₁₀r = 3.9 /2

log₁₀r = 1.95

r = 89.125 m

Therefore, the required distance is 89.125 m