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The atmospheric pressure p on an object decreases with increasing height. This pressure, measured in millimeters of mercury, is related to the height h (in kilometers) above sea level by the function p(h)=760e0.145hp(h)=760 e^{-0.145 h} Find the height of a mountain if the atmospheric pressure is 667 millimeters of mercury.


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Substitute 667667 to p(h)p(h) to obtain:

667=760e0.145h667760=760e0.145h760667760=e0.145h\begin{align*} 667&=760e^{-0.145h}\\ \frac{667}{760}&=\frac{760e^{-0.145h}}{760}\\ \frac{667}{760}&=e^{-0.145h}\\ \end{align*}

Use the rule ea=b    a=lnbe^a=b \implies a=\ln{b} to obtain:

0.145h=ln(667760)0.145h0.145=ln(667760)0.145h=ln(667760)0.145\begin{align*} -0.145h&=\ln{\left(\frac{667}{760}\right)}\\ \frac{-0.145h}{-0.145}&=\dfrac{\ln{\left(\frac{667}{760}\right)}}{-0.145}\\ h&=\dfrac{\ln{\left(\frac{667}{760}\right)}}{-0.145} \end{align*}

Use a calculator to obtain:

h=0.9001957749h0.90 km\begin{align*} h&=0.9001957749\\ h&\approx 0.90\text{ km} \end{align*}

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