Calculating Crowd Size at a Football Game

[Salty]

When a single person shouts at a football game, the sound intensity level at the center of the field is 60 dB. When all the people shout together the intensity level increases to 109 dB. Assuming that each person generates the same sound intensity at the center of the field, how many people are at the game?

 

[James R]

What's the relationship between sound intensity level and intensity? Do you know the relevant equation?

Hint: Sound intensity level is measured in decibels. Intensity is measured in Watts per square metre.

 

[thepatientmental]

To calculate the number of people at the football game, we can use the inverse square law, which states that sound intensity decreases as the distance from the source increases. In this scenario, we can assume that the person shouting at the center of the field is the source of the sound and the crowd is the receiver.

Using the inverse square law, we can set up the following equation:

I1 / I2 = (d2 / d1)^2

Where:
I1 = sound intensity level at the center of the field when a single person shouts (60 dB)
I2 = sound intensity level at the center of the field when all people shout (109 dB)
d1 = distance from the source (center of the field) to the receiver (single person)
d2 = distance from the source (center of the field) to the receiver (crowd)

Substituting the given values, we get:

60 / 109 = (d2 / d1)^2

Solving for d2 / d1, we get:

d2 / d1 = √(60 / 109) = 0.575

This means that the distance from the source to the receiver (crowd) is 0.575 times the distance from the source to the receiver (single person).

Now, we can assume that the single person is at the center of the field, which is equal to half the radius of the field. This means that the radius of the field is d1 / 2.

Substituting this in the equation above, we get:

d2 = 0.575 * (d1 / 2) = 0.2875 * d1

Since we know that the radius of the field is approximately 50 yards (150 feet), we can set up another equation:

d1 = 150 feet

Substituting this in the equation above, we get:

d2 = 0.2875 * 150 = 43.125 feet

Now, we can calculate the area of the field using the formula for the area of a circle:

A = π * r^2 = π * (43.125)^2 = 5852.54 square feet

Assuming that each person takes up approximately 4 square feet of space, we can divide the total area of the field by 4 to get the maximum number of people that can fit in the field:

5852.54 / 4