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Chapter 3: Problem 64

Halley's law states that the barometric pressure (in inches of mercury) at an altitude of \(x \mathrm{mi}\) above sea level is approximated by the equation $$ p(x)=29.92 e^{-0.2 x} \quad(x \geq 0) $$ If the barometric pressure as measured by a hot-air balloonist is 20 in. of mercury, what is the balloonist's altitude?

Short Answer

Expert verified
The hot-air balloonist's altitude is approximately 8.29 miles above sea level.

Step by step solution

01

We are given the equation for Halley's law, which relates the barometric pressure (p) to the altitude (x) in miles: \[p(x) = 29.92 e^{-0.2 x}\] We are also given that the barometric pressure as measured by a hot-air balloonist is 20 inches of mercury. #Step 2: Set the equation to the given pressure and solve for x#

We know the pressure is 20 inches of mercury, so we can set the equation to 20 and solve for x: \[20 = 29.92 e^{-0.2 x}\] #Step 3: Isolate the exponential term#
02

To isolate the exponential term, divide both sides by 29.92: \[\frac{20}{29.92} = e^{-0.2 x}\] #Step 4: Take the natural logarithm of both sides#

Now we will take the natural logarithm (ln) of both sides to remove the exponential term. Recall that ln(e^x) = x: \[\ln\left(\frac{20}{29.92}\right) = \ln\left(e^{-0.2 x}\right) \Rightarrow \ln\left(\frac{20}{29.92}\right) = -0.2x\] #Step 5: Solve for x#
03

Finally, solve for x by dividing both sides by -0.2: \[x = \frac{\ln\left(\frac{20}{29.92}\right)}{-0.2}\] Plug the natural logarithm values into a calculator and solve: \[x \approx 8.29\] #Step 6: Interpret the result#

The hot-air balloonist's altitude is approximately 8.29 miles above sea level.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Barometric Pressure Equation
Balloonists, hikers, pilots, and weather enthusiasts frequently rely on knowing the barometric pressure, which is the pressure exerted by the weight of the atmosphere. The barometric pressure equation is a mathematical relationship that describes how air pressure changes with altitude.

In Halley's Law, a specific barometric pressure equation is given as \( p(x) = 29.92 e^{-0.2x} \), where \( p(x) \) represents the pressure in inches of mercury at an altitude \( x \), in miles, above sea level. This equation is particularly useful for calculating the pressure at different altitudes, or in the reverse manner, finding an altitude when a pressure is known.

The equation reflects an inverse relationship; as the altitude increases, the barometric pressure decreases, and vice versa.
Exponential Functions
Exponential functions play a critical role in various scientific and financial calculations, describing quantities that change proportionally to their current value. They are written in the form \(y = ab^{x}\), where \(b\) is the base, \(x\) is the exponent, and \(a\) represents the coefficient.

In the context of the barometric pressure equation, the exponential term \(e^{-0.2x}\) dictates the rate at which pressure decreases with altitude. The base \(e\) is the mathematical constant approximately equal to 2.71828, and is the base of the natural logarithm. Bridging the concrete realm of measurements with the abstract world of mathematical constants, the exponential function elegantly captures the rapidly diminishing nature of atmospheric pressure with height.
Natural Logarithm Applications
The natural logarithm, typically denoted as \(ln\), is an essential mathematical tool used to unravel exponential functions. Its most notable application is solving equations where the variable is in an exponent. The natural logarithm has a special property that \(ln(e^x) = x\).

When dealing with the exponential decrease of pressure with altitude, as shown in Halley's Law, taking the natural logarithm of both sides enables us to solve for the altitude \(x\) given a particular pressure. This property helps in extracting the variable from an exponent, making it easier to solve complex equations that arise not only in physics and engineering but in growth models of economics and biology as well.

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