When I sing into a microphone, suppose I produce a sound intensity level (SIL) of 40 dB. If you can sing at 60 dB, what is the ratio of our produced sound intensities (yours divided by mine)?

To find the ratio of the produced sound intensities, you need to use the concept of sound intensity level (SIL) and its relationship with decibels (dB). The formula relating sound intensity (I) to SIL is as follows:

SIL = 10 * log10(I/I_0),

where I is the sound intensity being measured and I_0 is the reference intensity, which is usually taken as the quietest sound that a typical human ear can hear (approximately 10^(-12) W/m^2).

In this case, we know that your SIL is 60 dB and mine is 40 dB. Let's calculate the ratio of our sound intensities by using the formula:

60 dB = 10 * log10(I_your/I_0), (for your SIL)
40 dB = 10 * log10(I_mine/I_0). (for my SIL)

First, rearrange the equations to solve for the sound intensities:

I_your/I_0 = 10^(60/10),
I_mine/I_0 = 10^(40/10).

Now, divide the equation for your sound intensity by the equation for my sound intensity to obtain the ratio:

(I_your/I_0) / (I_mine/I_0) = (10^(60/10)) / (10^(40/10)).

Simplifying further:

(I_your/I_0) / (I_mine/I_0) = 10^((60-40) / 10) = 10^(20/10) = 10^2 = 100.

Therefore, the ratio of our produced sound intensities is 100:1, which means that your sound intensity is 100 times greater than mine.