1. **Stating the problem:** We have two earthquakes with magnitudes $M_1=6.5$ and $M_2=6.0$ on the Richter scale. We want to find: - The ratio $\frac{A_1}{A_2}$ of the amplitudes of the seismic waves. - The percentage change in energy released between the two earthquakes using the Gutenberg-Richter law. 2. **Formulas and rules:** - Richter magnitude formula: $$M = \log_{10}\left(\frac{A}{A_0}\right)$$ where $A$ is the amplitude and $A_0$ a reference amplitude. - Gutenberg-Richter energy formula: $$\log_{10}\left(\frac{E}{E_0}\right) = 1.5M + 4.8$$ where $E$ is energy and $E_0$ a reference energy. 3. **Finding the amplitude ratio $\frac{A_1}{A_2}$:** From the Richter formula: $$M_1 = \log_{10}\left(\frac{A_1}{A_0}\right), \quad M_2 = \log_{10}\left(\frac{A_2}{A_0}\right)$$ Subtracting: $$M_1 - M_2 = \log_{10}\left(\frac{A_1}{A_0}\right) - \log_{10}\left(\frac{A_2}{A_0}\right) = \log_{10}\left(\frac{A_1}{A_2}\right)$$ Therefore: $$\frac{A_1}{A_2} = 10^{M_1 - M_2} = 10^{6.5 - 6.0} = 10^{0.5}$$ Numerically: $$10^{0.5} = \sqrt{10} \approx 3.162$$ 4. **Finding the percentage change in energy:** Using Gutenberg-Richter: $$\log_{10}\left(\frac{E_1}{E_0}\right) = 1.5M_1 + 4.8$$ $$\log_{10}\left(\frac{E_2}{E_0}\right) = 1.5M_2 + 4.8$$ Subtracting: $$\log_{10}\left(\frac{E_1}{E_0}\right) - \log_{10}\left(\frac{E_2}{E_0}\right) = 1.5(M_1 - M_2)$$ Which gives: $$\log_{10}\left(\frac{E_1}{E_2}\right) = 1.5(6.5 - 6.0) = 1.5 \times 0.5 = 0.75$$ So: $$\frac{E_1}{E_2} = 10^{0.75}$$ Numerically: $$10^{0.75} = 10^{\frac{3}{4}} = \sqrt[4]{10^3} = \sqrt[4]{1000} \approx 5.623$$ 5. **Percentage increase in energy:** $$\text{Percentage increase} = \left(\frac{E_1}{E_2} - 1\right) \times 100 = (5.623 - 1) \times 100 = 462.3\%$$ **Final answers:** - Amplitude ratio: $$\frac{A_1}{A_2} \approx 3.162$$ - Energy increase: approximately $$462.3\%$$