The loudness of a sound can be measured in terms of watts of sonic energy P. Usually, however, it is measured with decibels.

Question

The loudness of a sound can be measured in terms of watts of sonic energy P. Usually, however, it is measured with decibels. The relationship between the decibel level L and the watts of sonic energy P is given by

L=(10/(ln10))ln(P/P₀₎

where P₀ is an extremely small constant (10^-12, although you do not need to know that to solve this problem.)

A jackhammer emits 1 watt of sonic energy when it is running. When it is turned off, it will take a moment to completely stop; as it slows down, it becomes quieter. Half a second after the jackhammer is turned off, it is emitting 0.25 watts of sonic energy, and that amount is decreasing by 0.75 watts/second. How is the decibel level changing at that moment?

Answer 1

What is wanted is dL/dt when P = 0.25 = 1/4 and dP/dt = -0.75 = -3/4

Let k=10/ln(10) , a constant.

L = kln(P/P₀) = kln(P)-kln(P₀) where kln(P₀) is just another constant and taking the derivative of a constant

is zero.

So,

dL/dt = (k/P) dP/dt

Just substitute in the values for k, P and dP/dt.

dL/dt = (10/ln(10)(4)(-3/4)≈-13.029 decibels/sec Note: 1/(1/4))=4