Air pressure may be represented as a function of height above the surface of the Earth as shown below:
[tex]$P(h)=P_o e^{-00012 h}$[/tex]
In this function, [tex]$P_0$[/tex] 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?
A. [tex]$P_o=.65 P_o e^{-.00012 h}$[/tex]
B. [tex]$65=h \cdot e^{-.00012}$[/tex]
C. [tex]$.65 P_o=P_o e^{-.00012 h}$[/tex]
D. [tex]$h=.65 e^{-0.0012}$[/tex]
Answer (2)
The problem provides a function for air pressure as a function of height: P ( h ) = P o e − 0.00012 h .
We want to find the height h when the air pressure is 65% of the air pressure at sea level, so P ( h ) = 0.65 P 0 .
Substitute 0.65 P 0 into the equation: 0.65 P 0 = P o e − 0.00012 h .
The correct equation is 0.65 P o = P o e − 0.00012 h .
Explanation
Understanding the Problem We are given the function P ( h ) = P o e − 0.00012 h , where P ( h ) is the air pressure at height h , and P 0 is the air pressure at sea level. We want to find the height h at which the air pressure is 65% of the air pressure at sea level. This means we want to find h such that P ( h ) = 0.65 P 0 .
Setting up the Equation To find the equation that will give us the desired height, we set P ( h ) equal to 0.65 P 0 in the given equation: 0.65 P 0 = P o e − 0.00012 h This equation represents the condition where the air pressure at height h is 65% of the air pressure at sea level.
Identifying the Correct Equation Now, we compare this equation with the given options: A. P o = .65 P o e − .00012 h B. 65 = h × e − .00012 C. .65 P o = P o e − .00012 h D. h = .65 e − 0.0012 Option C, 0.65 P o = P o e − 0.00012 h , matches the equation we derived.
Final Answer Therefore, the correct equation to find the height at which air pressure is 65% of the air pressure at sea level is: 0.65 P o = P o e − 0.00012 h
Examples
Understanding how air pressure changes with altitude is crucial in many real-world applications. For example, pilots need to know the air pressure at different altitudes to properly calibrate their instruments. Similarly, meteorologists use air pressure data to predict weather patterns. This equation helps to model and predict these changes, allowing for safer and more accurate predictions.
To find the height at which air pressure is 65% of sea level pressure, we set P ( h ) = 0.65 P 0 and derived the equation 0.65 P 0 = P 0 e − 0.00012 h . The correct option among the given choices is C. 0.65 P 0 = P 0 e − 0.00012 h .
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