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

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.