The problem asks to determine how many times louder a thunder clap is than a dog barking, given that the sound level of a dog barking is 83 dB and the sound level of a thunder clap is 102 dB.

1. Problem Description

The problem asks to determine how many times louder a thunder clap is than a dog barking, given that the sound level of a dog barking is 83 dB and the sound level of a thunder clap is 102 dB.

2. Solution Steps

The decibel scale is logarithmic. The difference in decibels between two sounds is related to the ratio of their intensities. Specifically, the difference in decibels ($dB$) is given by the formula:

$dB = 10 \log_{10}(\frac{I_2}{I_1})$

where $I_1$ and $I_2$ are the intensities of the two sounds.

Let $I_d$ be the intensity of the dog barking and $I_t$ be the intensity of the thunder clap. The sound level of the dog barking is 83 dB, and the sound level of the thunder clap is 102 dB. The difference in decibels is:

$102 - 83 = 19 \text{ dB}$

So, we have:

$19 = 10 \log_{10}(\frac{I_t}{I_d})$

Divide both sides by 10:

$1.9 = \log_{10}(\frac{I_t}{I_d})$

Now, we can rewrite this equation in exponential form:

$\frac{I_t}{I_d} = 10^{1.9}$

$\frac{I_t}{I_d} \approx 79.43$

Therefore, the thunder clap is approximately 79.43 times louder than the dog barking.

3. Final Answer

79.43