3. Earthquakes. The Richter Seale is used to measure the intensity of earthquakes using the formula \( R=\frac{2}{3}(\log E-11,8) \) where \( E \) is the energy released by the earthquake measured in ergs. a. The San Francisco earthquake of 1906 registered \( R=8,2 \) on the Richter seale, How many ergs of energy were released? b. In 1989 another San Francisco earthquake registered \( R=7, I \) on the Richter seale, How many times stronger (in terms of energy released) was the 1906 earthquake than this one? c. Ifone earthquake registers 2 points more than another on the Richter scale how many times more energy does it release?

**(a)** We are given \[ R = \frac{2}{3}(\log E - 11.8) \] with \( R = 8.2 \). Substitute and solve: \[ 8.2 = \frac{2}{3}(\log E - 11.8). \] Multiply both sides by \(\frac{3}{2}\): \[ \log E - 11.8 = \frac{3}{2} \times 8.2 = 12.3. \] Add 11.8 to both sides: \[ \log E = 11.8 + 12.3 = 24.1. \] Thus, the energy is \[ E = 10^{24.1} \text{ ergs}. \] **(b)** For the 1989 earthquake, the Richter reading is \( R = 7.1 \). Substitute into the formula: \[ 7.1 = \frac{2}{3}(\log E - 11.8). \] Multiply both sides by \(\frac{3}{2}\): \[ \log E - 11.8 = \frac{3}{2} \times 7.1 = 10.65. \] Add 11.8 to both sides: \[ \log E = 11.8 + 10.65 = 22.45. \] Thus, the energy released in 1989 is \[ E = 10^{22.45} \text{ ergs}. \] To find how many times stronger the 1906 earthquake was, compute the ratio: \[ \frac{E_{1906}}{E_{1989}} = \frac{10^{24.1}}{10^{22.45}} = 10^{24.1 - 22.45} = 10^{1.65}. \] Calculating \( 10^{1.65} \) gives approximately 44.7. Hence, the 1906 earthquake released about 44.7 times more energy than the 1989 earthquake. **(c)** If one earthquake registers 2 points more than another, let the energies be \( E_1 \) and \( E_2 \) with corresponding magnitudes \( R_1 \) and \( R_2 \) such that \[ R_1 - R_2 = 2. \] Since \[ R = \frac{2}{3}(\log E - 11.8), \] we have \[ R_1 - R_2 = \frac{2}{3}(\log E_1 - \log E_2) = \frac{2}{3} \log \left(\frac{E_1}{E_2}\right). \] Set up the equation: \[ 2 = \frac{2}{3} \log \left(\frac{E_1}{E_2}\right). \] Multiply both sides by \(\frac{3}{2}\): \[ \log \left(\frac{E_1}{E_2}\right) = 3. \] Thus, \[ \frac{E_1}{E_2} = 10^3 = 1000. \] An earthquake that is 2 points higher on the Richter scale releases 1000 times more energy.