Final answer:
The decibel rating of a sound that is 10,000 times more intense than the initial intensity is 40 dB higher than the initial level.
Step-by-step explanation:
The decibel rating of a sound with intensity that is 10,000 times the initial intensity can be found using the formula β (dB) = 10 log10(I/I0), where I is the intensity of the sound and I0 is the initial intensity. Since each factor of 10 in intensity corresponds to 10 dB increase, an increase of 10,000 times is equivalent to four factors of 10 (10⁴). Therefore, the sound intensity level would be 40 dB higher than the initial intensity level.
If the initial intensity level were, for instance, the threshold of hearing at 0 dB, then the new intensity level would be 40 dB (0 dB + 40 dB from the four factors of ten increase).