Rational Equations

Question

Calvin finds that the volume of laughter in response to his jokes (v) varies directly with the square of the number of people present (p), but inversely to the length of time to tell the joke (t). the volume is 360 db when 10 people listen to a 5 second joke. how loud will the response be when 16 people listen to a 24 second joke?

Answer 1

V relates to P²/T  where V = volume in decibels, P = # people present, T = time

 

We get 360 db relating to 10²/5 = 100/5 =20

360 relates to 20 by a factor of 18(P²/T)

 

If I read this correctly then we find  V = 18(P²/T)

 

18(16²/24) = 18(10 2/3) = 192 db

Answer 2

In these types of questions, it is a good idea to set up some proportions to help define relationship between variables.

 

v ∝ p²

 

v ∝ 1 / t

 

 

Using these proportions, we can establish a simple formula to define v.

 

v = (Kp²) / t

 

where K is a constant.

 

 

We must find the value of K using the initial conditions:  v=360 , p=10, and t=5. 

 

K = (vt) / p²

 

K = (360 db * 5 sec) / 10²

 

K = (1800 db sec) / 100

 

K = 18 db sec

 

 

Next, we substitute this value of K along with the other values to find the volume of response: K=18 , p=16 , and  t=24

 

v = (Kp²) / t

 

v = (18 db sec * 16²) / 24 sec

 

v = (4608 db sec) / 24 sec

 

v = 192 db

 

 

The volume of the response is 192 db. 

 

Notice that units were included in the calculations.  This helps ensure that the work is calculation are being done correctly.