Step 1 :The problem is asking for the height at which the air pressure is 65% of the air pressure at sea level. This means we need to set \(P(h)\) equal to \(0.65P_{0}\) and solve for \(h\). The equation that represents this is \(0.65 P_{0}=P_{0} e^{-0.00012 h}\).
Step 2 :We can solve this equation for \(h\) by dividing both sides by \(P_{0}\) and then taking the natural logarithm of both sides.
Step 3 :Let \(P = 0.65\)
Step 4 :Solving the equation gives \(h = 3589.857634103785\)
Step 5 :The height at which air pressure is 65% of the air pressure at sea level is approximately \(\boxed{3589.86}\) meters.