$$ \leqslant \quad \begin{array}{l} ext { Use the earthquake magnitude scale to answer the following question. Compare the energy of a magnitude } 8 \ ext { earthquake to an explosive device that released } 13 imes 10^{18} ext { joules of energy. }\end{array} $$

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$$ \leqslant \quad \begin{array}{l}	ext { Use the earthquake magnitude scale to answer the following question. Compare the energy of a magnitude } 8 \ 	ext { earthquake to an explosive device that released } 13 	imes 10^{18} 	ext { joules of energy. }\end{array} $$

UpStudy AI · Accepted Answer

To compare the energy of a magnitude 8 earthquake to an explosive device that released $$ 13 \times 10^{18} $$ joules of energy, we can use the formula for the energy released by earthquakes on the Richter scale.

The energy $$ E $$ released by an earthquake of magnitude $$ M $$ can be approximated by the formula:

$$
E = 10^{(1.5M + 4.8)}
$$

### Step 1: Calculate the energy of a magnitude 8 earthquake

Substituting $$ M = 8 $$ into the formula:

$$
E = 10^{(1.5 \times 8 + 4.8)}
$$

Calculating the exponent:

$$
1.5 \times 8 = 12
$$
$$
12 + 4.8 = 16.8
$$

Now, we can calculate $$ E $$:

$$
E = 10^{16.8}
$$

### Step 2: Compare the energies

Now we need to compare $$ 10^{16.8} $$ joules with $$ 13 \times 10^{18} $$ joules.

To do this, we can express $$ 10^{16.8} $$ in a similar format:

$$
10^{16.8} = 10^{16.8} \text{ joules}
$$

### Step 3: Calculate $$ 10^{16.8} $$

Now, let's calculate $$ 10^{16.8} $$:

$$
10^{16.8} \approx 6.30957 \times 10^{16} \text{ joules}
$$

### Step 4: Compare $$ 6.30957 \times 10^{16} $$ joules with $$ 13 \times 10^{18} $$ joules

Now we can compare:

$$
6.30957 \times 10^{16} \text{ joules} \quad \text{and} \quad 13 \times 10^{18} \text{ joules}
$$

### Step 5: Calculate the ratio

To find out how many times more energy the explosive device has compared to the earthquake, we can calculate the ratio:

$$
\text{Ratio} = \frac{13 \times 10^{18}}{6.30957 \times 10^{16}}
$$

Calculating this ratio:

$$
\text{Ratio} \approx \frac{13}{6.30957} \times 10^{2} \approx 2.063 \times 10^{2} \approx 206.3
$$

### Conclusion

The explosive device that released $$ 13 \times 10^{18} $$ joules of energy is approximately $$ 206.3 $$ times more powerful than a magnitude 8 earthquake in terms of energy released.
A magnitude 8 earthquake releases about $$ 6.3 \times 10^{16} $$ joules of energy, while an explosive device releasing $$ 13 \times 10^{18} $$ joules is roughly 206 times more powerful.

\( \leqslant \quad \begin{array}{l}\text { Use the earthquake magnitude scale to answer the following question. Compare the energy of a magnitude } 8 \\ \text { earthquake to an explosive device that released } 13 \times 10^{18} \text { joules of energy. }\end{array} \)

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