Problem 130 - A Graphical Approach to College Algebra

Question

Solve each problem. Air Density As the altitude increases, air becomes thinner, or less dense. An approximation of the density \(d\) of air at an altitude of \(x\) meters above sea level is $$d(x)=\left(3.32 \times 10^{-9}\right) x^{2}-\left(1.14 \times 10^{-4}\right) x+1.22$$ The output is the density of air in kilograms per cubic meter. The domain of \(d\) is \(0 \leq x \leq 10,000 .\) (Source: A. Miller and J. Thompson, Elements of Meteorology.) (a) Denver is sometimes referred to as the mile-high city. Compare the density of air at sea level and in Denver. (Hint: \(1 \mathrm{ft} \approx 0.305 \mathrm{m}\) ) (b) Determine the altitudes where the density is greater than 1 kilogram per cubic meter.

Step-by-Step Solution

Answer

Sea level density: 1.22 kg/m³; Denver density: 0.95 kg/m³. Density > 1 kg/m³ for \(x \in [0, 340.8) \cup (9600.2, 10000]\).

Step 1: Understand the Function

The function given is a quadratic function \(d(x)=\left(3.32 \times 10^{-9}\right) x^{2}-\left(1.14 \times 10^{-4}\right)x+1.22\). It approximates air density in kg/m³ at an altitude \(x\) meters. The products \(3.32 \times 10^{-9}x^2\) and \(1.14 \times 10^{-4}x\) show the effect of altitude on density.

Step 2: Convert Mile-High to Meters

To find Denver's altitude, convert 1 mile to meters using the conversion \(1 \, \text{mile} = 5280 \, \text{feet}\), and \(1 \, \text{ft} \approx 0.305 \, \text{m}\). This results in \(5280 \times 0.305 = 1610.4 \, \text{meters}\).

Step 3: Calculate Air Density at Sea Level

Substitute \(x = 0\) (sea level altitude) into the function: \(d(0) = (3.32 \times 10^{-9})(0)^{2} - (1.14 \times 10^{-4})(0) + 1.22 = 1.22\, \text{kg/m}^3\).

Step 4: Calculate Air Density in Denver

Substitute \(x = 1610.4\) meters into the function: \[d(1610.4) = (3.32 \times 10^{-9})(1610.4)^{2} - (1.14 \times 10^{-4})(1610.4) + 1.22\]Calculate: \[d(1610.4) \approx 0.95\, \text{kg/m}^3\].

Step 5: Set Up Inequality for Density Greater Than 1

To find the altitude where density is greater than 1 kg/m³, solve the inequality: \[(3.32 \times 10^{-9})x^2 - (1.14 \times 10^{-4})x + 1.22 > 1\].

Step 6: Solve the Inequality

Reorder the inequality: \[(3.32 \times 10^{-9})x^2 - (1.14 \times 10^{-4})x + 0.22 > 0\]. Solve this quadratic inequality using the quadratic formula where \(a = 3.32 \times 10^{-9}, b = -1.14 \times 10^{-4}, c = 0.22\). Find the roots of the equation \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Calculate to find critical points around \(x \approx 340.8\) and \(x \approx 9600.2\). Since the parabola opens upwards, test intervals: density > 1 for \([0, 340.8) \cup (9600.2, 10000]\).

Step 7: Conclusion

Air is denser at sea level with \(1.22\, \text{kg/m}^3\) than in Denver (0.95 kg/m³). Density exceeds 1 kg/m³ at altitudes \(0 \leq x < 340.8\) meters and \(9600.2 < x \leq 10000\) meters.

Key Concepts

Quadratic Function

A quadratic function is a polynomial expression of degree two, usually written in the form \(ax^2 + bx + c\). This specific type of function graphs as a parabola. For the air density function given, \(d(x) = (3.32 \times 10^{-9}) x^{2} - (1.14 \times 10^{-4}) x + 1.22\), each term represents a part of the relationship between altitude \(x\) and air density \(d\).

This function shows how air density changes with altitude:

• The coefficient \(3.32 \times 10^{-9}\) attached to \(x^2\) indicates how rapidly the density changes with respect to altitude squared.
• The linear term \(-1.14 \times 10^{-4}x\) shows the linear decrease in density with increasing altitude.
• The constant \(1.22\) represents the air density at sea level, as when \(x = 0\), the function returns this value.

By understanding each component of the quadratic function, students can better predict and comprehend the changes in air density as elevation increases.

Altitude Conversion

Altitude conversion is vital when you need to understand metrics like altitude in different unit systems. For the given problem, we convert Denver's altitude from miles to meters to use the quadratic function to calculate air density.

To make this conversion:

• Begin with the conversion from miles to feet: \(1 \text{ mile} = 5280 \text{ feet}\).
• Then convert feet to meters: \(1 \text{ foot} \approx 0.305 \text{ meters}\).

For Denver, approximately one mile translates to about 1610.4 meters once we complete these conversions. Converting between units is not only crucial for mathematics. It is also important in fields like meteorology, where altitude affects weather conditions and calculations like air density.

Inequality Solving

Solving inequalities involves finding the range of values that satisfy the inequality statement. In this exercise, we are tasked with finding altitudes where air density exceeds 1 kg/m³.

To solve the inequality \((3.32 \times 10^{-9})x^2 - (1.14 \times 10^{-4})x + 1.22 > 1\), we transformed it to \((3.32 \times 10^{-9})x^2 - (1.14 \times 10^{-4})x + 0.22 > 0\).
Using the quadratic formula, we find the critical points that help define our solution:

• The formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) gives us these roots.
• Calculate these to find approximately \(x = 340.8\) and \(x = 9600.2\).

Knowing that the parabola opens upwards, we observe that the function is positive in specific intervals. Therefore, air density will exceed 1 kg/m³ at altitudes \([0, 340.8) \cup (9600.2, 10000]\). This approach not only resolves the current problem but also applies to many scientific and economic models using inequalities.

Meteorology

Meteorology is the science of the atmosphere, encompassing everything from weather forecasting to climate change studies. Understanding air density is crucial in this field as it influences various meteorological phenomena such as air pressure, wind, and temperature.

Air density, often calculated with functions like the one in the exercise, varies with altitude. Such changes help meteorologists predict weather patterns and influence aviation, agriculture, and even everyday activities. For instance, air density affects:

• Weather forecasting: Denser air can indicate lower altitudes or cold air masses, impacting storm prediction.
• Aviation: Pilots must know air density to calculate lift and fuel efficiency.
• Athletics: Air density affects sports played outdoors, altering ball trajectory and the performance of athletes.

Thus, while the mathematical calculations might seem abstract at first, they have significant real-world applications, making them essential for students interested in weather and atmospheric sciences.