Let's solve this problem step-by-step.
Given the equation for mean sustained wind velocity:
[tex]\[ v = 6.3 \sqrt{1013 - \rho} \][/tex]
We need to find the air pressure [tex]\( \rho \)[/tex] when the mean sustained wind velocity [tex]\( v \)[/tex] is 64 meters per second.
1. Substitute [tex]\( v = 64 \)[/tex] into the equation:
[tex]\[ 64 = 6.3 \sqrt{1013 - \rho} \][/tex]
2. Isolate the square root term by dividing both sides of the equation by 6.3:
[tex]\[ \frac{64}{6.3} = \sqrt{1013 - \rho} \][/tex]
3. Calculate the left-hand side:
[tex]\[ \frac{64}{6.3} \approx 10.1587 \][/tex]
4. Now, square both sides to eliminate the square root:
[tex]\[ \left(10.1587\right)^2 = 1013 - \rho \][/tex]
[tex]\[ 103.202 \approx 1013 - \rho \][/tex]
5. Solve for [tex]\( \rho \)[/tex]:
[tex]\[ \rho = 1013 - 103.202 \][/tex]
[tex]\[ \rho \approx 909.8 \][/tex]
So the air pressure at the center of the hurricane when the mean sustained wind velocity is 64 meters per second is approximately 910 millibars.
Among the given options, the closest value is:
- 103 millibars
- 194 millibars
- 363 millibars
- [tex]\( \boxed{910 \text{ millibars}} \)[/tex]
