To determine the air pressure [tex]\( p \)[/tex] at the center of the hurricane when the mean sustained wind velocity [tex]\( v \)[/tex] is 64 meters per second, we start with the given equation:
[tex]\[ v = 6.3 \sqrt{1013 - p} \][/tex]
We are provided that [tex]\( v = 64 \)[/tex]. Substituting this value into the equation, we get:
[tex]\[ 64 = 6.3 \sqrt{1013 - p} \][/tex]
To solve for [tex]\( p \)[/tex], follow these steps:
1. Isolate the square root term:
[tex]\[ \frac{64}{6.3} = \sqrt{1013 - p} \][/tex]
2. Calculate the left-hand side:
[tex]\[ \frac{64}{6.3} \approx 10.1587 \][/tex]
3. Square both sides to eliminate the square root:
[tex]\[ (10.1587)^2 = 1013 - p \][/tex]
[tex]\[ 103.1998 \approx 1013 - p \][/tex]
4. Isolate [tex]\( p \)[/tex]:
[tex]\[ p = 1013 - 103.1998 \][/tex]
[tex]\[ p \approx 909.8002 \][/tex]
The calculated air pressure [tex]\( p \approx 909.8002 \)[/tex] millibars is very close to one of the options given, 910 millibars. Therefore, the approximate air pressure at the center of the hurricane when the mean sustained wind velocity is 64 meters per second is:
[tex]\[ \boxed{910 \text{ millibars}} \][/tex]
