Answer :
To determine the approximate air pressure at the center of a hurricane when the mean sustained wind velocity is 64 meters per second, we start by using the given equation [tex]\( v = 6.3 \sqrt{1013 - p} \)[/tex], where [tex]\( v \)[/tex] is the wind velocity in meters per second and [tex]\( p \)[/tex] is the air pressure in millibars.
### Step-by-Step Solution:
1. Substitute the given wind velocity into the equation:
[tex]\[ 64 = 6.3 \sqrt{1013 - p} \][/tex]
2. Isolate the square root term by dividing both sides of the equation by 6.3:
[tex]\[ \frac{64}{6.3} = \sqrt{1013 - p} \][/tex]
3. Simplify the left side:
[tex]\[ \frac{64}{6.3} \approx 10.16 \][/tex]
So:
[tex]\[ 10.16 \approx \sqrt{1013 - p} \][/tex]
4. Square both sides of the equation to eliminate the square root:
[tex]\[ (10.16)^2 = 1013 - p \][/tex]
[tex]\[ 103.23 \approx 1013 - p \][/tex]
5. Isolate [tex]\( p \)[/tex] by subtracting the squared value from 1013:
[tex]\[ p = 1013 - 103.23 \][/tex]
[tex]\[ p \approx 909.77 \][/tex]
6. Round the result to the nearest whole number:
[tex]\[ p \approx 910 \][/tex]
Therefore, the approximate air pressure at the center of the hurricane when the mean sustained wind velocity is 64 meters per second is 910 millibars.
[tex]\[ \boxed{910 \text{ millibars}} \][/tex]
### Step-by-Step Solution:
1. Substitute the given wind velocity into the equation:
[tex]\[ 64 = 6.3 \sqrt{1013 - p} \][/tex]
2. Isolate the square root term by dividing both sides of the equation by 6.3:
[tex]\[ \frac{64}{6.3} = \sqrt{1013 - p} \][/tex]
3. Simplify the left side:
[tex]\[ \frac{64}{6.3} \approx 10.16 \][/tex]
So:
[tex]\[ 10.16 \approx \sqrt{1013 - p} \][/tex]
4. Square both sides of the equation to eliminate the square root:
[tex]\[ (10.16)^2 = 1013 - p \][/tex]
[tex]\[ 103.23 \approx 1013 - p \][/tex]
5. Isolate [tex]\( p \)[/tex] by subtracting the squared value from 1013:
[tex]\[ p = 1013 - 103.23 \][/tex]
[tex]\[ p \approx 909.77 \][/tex]
6. Round the result to the nearest whole number:
[tex]\[ p \approx 910 \][/tex]
Therefore, the approximate air pressure at the center of the hurricane when the mean sustained wind velocity is 64 meters per second is 910 millibars.
[tex]\[ \boxed{910 \text{ millibars}} \][/tex]
