Question

The air pressure in the neighborhood of Mount Wolf is given by the equation p(x, y, z) = 38e⁽⁻⁷ˣ² ⁻ ⁴ʸ² ⁻ ²ᶻ⁾. If u² + v² ≤ 10,000, where distance is measured in meters, what is the maximum value of p(x, y, z)?

Answer 1

The maximum value of the air pressure function p(x, y, z) is found to be 38 pascals, occurring at the point where x = 0, y = 0, and z = 0, within the constraint u² + v² ≤ 10,000 meters.

Step-by-step explanation:

The maximum value of the air pressure function p(x, y, z) = 38e⁡(−7x² − 4y² − 2z) given the constraint u² + v² ≤ 10,000, where distance is measured in meters, occurs at the highest point where u and v are at their maximum values within the given constraint.

To find this maximum, we can observe that the function p is decreasing with respect to the square of the distance since there are negative coefficients for x², y², and z in the exponent of e.

Therefore, the function reaches its maximum when x = 0, y = 0, and implicit from the equation z = 0, as it provides the lowest value for the exponent (which is 0 in this case) and thus the highest value for the pressure function. Therefore, the maximum value of the function is 38e0 = 38 pascals.