1. **State the problem:** We are given the sea level pressure 500 km from the hurricane eye as 1000 hPa and the central pressure as 980 hPa. We need to find the wind speed $V$ using the formula $$V = \frac{1}{\rho f \Delta x} \Delta p$$ and then determine the Saffir-Simpson category of the hurricane. Next, we find how much the central pressure must drop for the hurricane to strengthen to Category 4. 2. **Given values:** - $\rho = 1.20$ kg/m$^3$ (air density) - $f = 8.34 \times 10^{-5}$ s$^{-1}$ (Coriolis parameter) - $\Delta x = 500$ km $= 500,000$ m (distance from eye) - Pressure at 500 km: $p_{outer} = 1000$ hPa - Initial central pressure: $p_{center} = 980$ hPa 3. **Convert pressure difference to Pascals:** Pressure difference $$\Delta p = p_{outer} - p_{center} = 1000 - 980 = 20 \text{ hPa}$$ Since 1 hPa = 100 Pa, $$\Delta p = 20 \times 100 = 2000 \text{ Pa}$$ 4. **Calculate wind speed $V$ using the formula:** $$V = \frac{1}{\rho f \Delta x} \Delta p = \frac{1}{1.20 \times 8.34 \times 10^{-5} \times 500,000} \times 2000$$ Calculate denominator: $$1.20 \times 8.34 \times 10^{-5} \times 500,000 = 1.20 \times 8.34 \times 50 = 1.20 \times 417 = 500.4$$ So, $$V = \frac{2000}{500.4} \approx 3.996 \text{ m/s}$$ 5. **Determine Saffir-Simpson category:** Wind speed $\approx 4$ m/s is very low, so the hurricane is below Category 1. 6. **Find wind speed for Category 4 hurricane:** Category 4 corresponds to wind speeds between 58 and 70 m/s. We use the lower bound 58 m/s to find the required pressure drop. 7. **Calculate required pressure difference $\Delta p_{new}$ for $V=58$ m/s:** Rearranged formula: $$\Delta p_{new} = V \times \rho f \Delta x = 58 \times 1.20 \times 8.34 \times 10^{-5} \times 500,000$$ Calculate: $$1.20 \times 8.34 \times 10^{-5} \times 500,000 = 500.4$$ So, $$\Delta p_{new} = 58 \times 500.4 = 29,023.2 \text{ Pa}$$ Convert to hPa: $$29,023.2 / 100 = 290.23 \text{ hPa}$$ 8. **Calculate new central pressure:** $$p_{center,new} = p_{outer} - \Delta p_{new} = 1000 - 290.23 = 709.77 \text{ hPa}$$ 9. **Calculate pressure drop from initial central pressure:** $$\text{Pressure drop} = 980 - 709.77 = 270.23 \text{ hPa}$$ **Final answers:** - Initial wind speed $\approx 4$ m/s, hurricane below Category 1. - To strengthen to Category 4, central pressure must drop by approximately 270 hPa to about 710 hPa.