Problem 79
Question
The height \(h\) above sea level (in meters) is related to air temperature \(t\) (in degrees Celsius), the atmospheric pressure \(p\) (in centimeters of mercury at height \(h\) ), and the atmospheric pressure \(c\) at sea level by $$ h=(30 t+8000) \ln (c / p) $$ If the pressure at the top of Mount Rainier is 44 centimeters on a day when sea level pressure is 75.126 centimeters and the temperature is \(7^{\circ} \mathrm{C},\) what is the height of Mount Rainier?
Step-by-Step Solution
Verified Answer
Answer: The estimated height of Mount Rainier is approximately 4392.35 meters above sea level.
1Step 1: Identify given values
From the problem, we are given the following values:
$$
t = 7 ^\circ \mathrm{C}
$$
$$
p = 44 \mathrm{~cm}
$$
$$
c = 75.126 \mathrm{~cm}
$$
2Step 2: Substitute values into the formula
We will now substitute these values into the given formula for \(h\):
$$
h = (30 \times t + 8000) \ln(c/p)
$$
3Step 3: Plug in the given values
Plug in the given values of \(t, p,\) and \(c\) into the formula:
$$
h = (30 \times 7 + 8000) \ln(75.126 / 44)
$$
4Step 4: Simplify and solve
Now, we will simplify and solve for \(h\):
$$
h = (210 + 8000) \ln(75.126 / 44) \approx (8210) \ln(1.708)
$$
$$
h \approx 8210 \times 0.535
$$
$$
h \approx 4392.35
$$
Thus, the estimated height of Mount Rainier is approximately 4392.35 meters above sea level.
Key Concepts
Atmospheric PressureNatural LogarithmAir Temperature Relation
Atmospheric Pressure
Atmospheric pressure is the force per unit area exerted on a surface by the weight of air above that surface in the atmosphere of Earth. It varies with location and time, as it is affected by factors such as temperature, altitude, and weather patterns.
Understanding atmospheric pressure is crucial when determining the height above sea level, as there is a predictable decrease in pressure with an increase in altitude. This requires precise measurement because a small error in pressure reading can lead to a significant error in calculating elevation.
In our example with Mount Rainier, atmospheric pressure measurements at the mountain's peak and at sea level played a key role in determining the mountain's height. The pressure differential, observed with a barometer, gives us essential data for the calculation. It's also important to note that standard atmospheric pressure at sea level is usually about 76 centimeters of mercury (cmHg), but this value can vary depending on weather conditions and more.
Understanding atmospheric pressure is crucial when determining the height above sea level, as there is a predictable decrease in pressure with an increase in altitude. This requires precise measurement because a small error in pressure reading can lead to a significant error in calculating elevation.
In our example with Mount Rainier, atmospheric pressure measurements at the mountain's peak and at sea level played a key role in determining the mountain's height. The pressure differential, observed with a barometer, gives us essential data for the calculation. It's also important to note that standard atmospheric pressure at sea level is usually about 76 centimeters of mercury (cmHg), but this value can vary depending on weather conditions and more.
Natural Logarithm
The natural logarithm, denoted as \(ln\), is a mathematical function that is the inverse of the exponential function. Natural logarithms are based on the number 'e' (approximately 2.71828), which is an irrational and transcendental number with important properties in calculus.
In practice, \(ln\) is used to solve equations in which a variable is an exponent, such as in the process of finding the height above sea level. It is also a useful tool for analyzing growth processes, compounding, and complex financial calculations. For students going through these calculations, being comfortable with using natural logarithms is crucial as it simplifies the process of manipulating exponential relationships.
In the height calculation exercise, we used the natural logarithm to understand the relationship between the atmospheric pressures at different heights. The ratio of the pressures is placed inside the logarithm to linearize the exponential relationship, enabling us to solve for the height easily.
In practice, \(ln\) is used to solve equations in which a variable is an exponent, such as in the process of finding the height above sea level. It is also a useful tool for analyzing growth processes, compounding, and complex financial calculations. For students going through these calculations, being comfortable with using natural logarithms is crucial as it simplifies the process of manipulating exponential relationships.
In the height calculation exercise, we used the natural logarithm to understand the relationship between the atmospheric pressures at different heights. The ratio of the pressures is placed inside the logarithm to linearize the exponential relationship, enabling us to solve for the height easily.
Air Temperature Relation
The correlation between air temperature and height above sea level is less direct but no less important. As one ascends into the atmosphere, air temperature generally decreases. The temperature can influence the density and pressure of the air, which in turn, affects calculations related to height.
The given formula in the exercise subtly incorporates temperature as a modifier of the pressure-height relationship. The term \(30t\) in the formula accounts for the change in pressure with temperature, and this linear approximation is common in environmental physics for moderate temperature ranges and altitudes.
For instance, in Mount Rainier's height calculation, the air temperature affected the atmospheric pressure, which was then used to determine the height above sea level. A keen understanding of this relationship is paramount when working with altitudes and atmospheric conditions, such as in meteorology, aeronautics, and even mountain climbing.
The given formula in the exercise subtly incorporates temperature as a modifier of the pressure-height relationship. The term \(30t\) in the formula accounts for the change in pressure with temperature, and this linear approximation is common in environmental physics for moderate temperature ranges and altitudes.
For instance, in Mount Rainier's height calculation, the air temperature affected the atmospheric pressure, which was then used to determine the height above sea level. A keen understanding of this relationship is paramount when working with altitudes and atmospheric conditions, such as in meteorology, aeronautics, and even mountain climbing.
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