Question

At a state championship High School football game, the intensity level of the shout of a single person in the stands at the center of the field is 48.1 dB. What would be the intensity level at the center of the field if all 4841 fans at the game shouted from roughly the same distance away from that center point

Answer 1

The value is
`\beta_f = 84.95 \ dB`

Step-by-step explanation:

From the question we are told that

The intensity level of the shout of a single person is
`\beta = 48.1 \ dB`

The number of fans is
`n = 4841`

Gnerally intensity level is mathematically represented as

`\beta = 10 log * (I)/(I_o )`

Here
`I_o` is the minimum intensity of sound human ear can pick and the value is

`I_o = 1 * 10^(-12) \ W/m ^2`

when
`\beta = 48.1 \ dB`

`48.1 = 10 log * (I)/( 1 * 10^(-12))`

=>
`4.81 = log ( ( I)/( 1 * 10^(-12)) )`

taking antilog of both sides

`64565.42 = (I)/( 1 *10^(-12))`

=>
`I = 6.457 *10^(-8) \ W/m^2`

Generally the intensity for the whole fans is mathematically represented as

`I_f = n * I`

=>
`I_f = 4841 * 6.457 *10^(-8 )`

=>
`I_f = 0.0003126 \ W/m^2`

Gnerally the intensity level for the whole fans is mathematically represented as

`\beta_f = 10 log [ (I_f )/(I_o ) ]`

=>
`\beta_f = 10 log [ ( 0.0003126 )/(1*10^(-12))`

=>
`\beta_f = 84.95 \ dB`