Air pressure may be represented as a function of height above the surface of the Earth as shown below.
\[
P(h)=P_{0} e^{-0.0012 h}
\]
In this function, $P_{0}$ is air pressure at sea level, and $h$ is measured in meters. Which of the following equations will find the height at which air pressure is $65 \%$ of the air pressure at sea level?
$P_{0}=0.65 P_{0} e^{-0.00012 n}$
$h=0.65 e^{-0.00012}$
$0.65=h \cdot e^{-0.00012}$
$0.65 P_{0}=P_{0} e^{-0.00012 h}$
Answer
The height at which air pressure is 65% of the air pressure at sea level is approximately \(\boxed{3589.86}\) meters.
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.
