To find the air pressure at the center of a hurricane when the mean sustained wind velocity [tex]\( v \)[/tex] is 64 meters per second, we need to solve the equation:
[tex]\[ v = 6.3 \sqrt{1013 - p} \][/tex]
where [tex]\( p \)[/tex] is the air pressure in millibars.
Given:
[tex]\[ v = 64 \text{ meters per second} \][/tex]
We start by rearranging the equation to isolate the square root term:
[tex]\[ 64 = 6.3 \sqrt{1013 - p} \][/tex]
Next, we solve for [tex]\( \sqrt{1013 - p} \)[/tex] by dividing both sides of the equation by 6.3:
[tex]\[ \sqrt{1013 - p} = \frac{64}{6.3} \][/tex]
Calculating the right-hand side:
[tex]\[ \frac{64}{6.3} \approx 10.1587 \][/tex]
Now, we square both sides to eliminate the square root:
[tex]\[ 1013 - p = (10.1587)^2 \][/tex]
Calculating the square of 10.1587:
[tex]\[ (10.1587)^2 \approx 103.1998 \][/tex]
Next, we solve for [tex]\( p \)[/tex] by isolating it on one side of the equation:
[tex]\[ p = 1013 - 103.1998 \][/tex]
Performing the subtraction:
[tex]\[ p \approx 909.8002 \text{ millibars} \][/tex]
So, the approximate air pressure at the center of the hurricane is 909.8002 millibars.
Now, we compare this value with the provided options:
- 103 millibars
- 194 millibars
- 363 millibars
- 910 millibars
The closest value to 909.8002 millibars is 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:
910 millibars.
