Question

question 20 of 20
air pressure may be represented as a function of height (in meters) above the surface of the earth, as shown below.
p(h)=p₀·e⁻⁰.⁰⁰⁰¹²h
in this function, p₀ is the air pressure at the surface of the earth, and h is the height above the surface of the earth, measured in meters. at what height will the air pressure equal 50% of the air pressure at the surface of the earth?
a. 5776.2 m
b. 0.59 m
c. 4166.7 m
d. 2148.9 m

Explanation:

Step1: Set up the equation

We want $P(h)=0.5P_0$. Substitute into the given formula $P(h)=P_0\cdot e^{- 0.00012h}$, we get $0.5P_0=P_0\cdot e^{-0.00012h}$. Since $P_0
eq0$, we can divide both sides by $P_0$ to obtain $0.5 = e^{-0.00012h}$.

Step2: Take the natural - logarithm of both sides

Taking the natural - logarithm of both sides of the equation $0.5 = e^{-0.00012h}$, we have $\ln(0.5)=\ln(e^{-0.00012h})$. According to the property of logarithms $\ln(e^x)=x$, the right - hand side simplifies to $-0.00012h$. So, $\ln(0.5)=-0.00012h$.

Step3: Solve for $h$

We know that $\ln(0.5)\approx - 0.6931$. Then, $h=\frac{\ln(0.5)}{-0.00012}=\frac{-0.6931}{-0.00012}\approx5776.2$ m.

Answer:

A. 5776.2 m