Problem 68 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

Density at sea level is 1.22 kg/m³, and 1.07 kg/m³ in Denver. Density > 1 kg/m³ from 0 to ~689.4 m and ~9794.8 m.

Step 1: Convert Mile to Meters

To find the altitude of Denver in meters, convert one mile to meters using the given conversion factor. Since 1 ft \(\approx\) 0.305 m, there are 5280 feet in a mile, thus 1 mile is approximately \(5280 \times 0.305 = 1609.44\) m. Therefore, Denver is about 1609.44 meters above sea level.

Step 2: Calculate Density at Sea Level

Substitute \(x = 0\) into the function \(d(x)\) to find the density at sea level. This gives: \[d(0) = (3.32 \times 10^{-9}) \times 0^2 - (1.14 \times 10^{-4}) \times 0 + 1.22 = 1.22 \, \text{kg/m}^3\]Thus, the density of air at sea level is 1.22 kg/m³.

Step 3: Calculate Density at Denver's Altitude

Substitute \(x = 1609.44\) m into the function \(d(x)\) to find the density at Denver's altitude. This gives: \[d(1609.44) = (3.32 \times 10^{-9}) \times 1609.44^2 - (1.14 \times 10^{-4}) \times 1609.44 + 1.22\]Calculate this using a calculator or software to get approximately 1.07 kg/m³. Thus, the density of air in Denver is approximately 1.07 kg/m³.

Step 4: Set Up Inequality for Part (b)

To find the altitudes where the density is greater than 1 kg/m³, solve the inequality:\[d(x) > 1 \ (3.32 \times 10^{-9}) x^2 - (1.14 \times 10^{-4}) x + 1.22 > 1\] Rearrange the inequality to:\[(3.32 \times 10^{-9}) x^2 - (1.14 \times 10^{-4}) x + 0.22 > 0\]

Step 5: Solve the Quadratic Inequality

Solve the quadratic inequality \((3.32 \times 10^{-9}) x^2 - (1.14 \times 10^{-4}) x + 0.22 > 0\). Using the quadratic formula, solve for \(x\):\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where \(a = 3.32 \times 10^{-9}\), \(b = -1.14 \times 10^{-4}\), and \(c = 0.22\). Calculate the discriminant, \(b^2 - 4ac = 1.2996 \times 10^{-8} - 2.9264 \times 10^{-9} = 1.007936 \times 10^{-8}\), and solve for \(x\) using the quadratic formula. The approximate solutions are \(x \approx 689.4\) m and \(x \approx 9794.8\) m. Hence, the air density is greater than 1 kg/m³ for altitudes \( 0 \leq x < 689.4\) m or \(689.4 < x < 9794.8\) m.

Key Concepts

Altitude Calculation

Altitude calculation helps us understand changes that occur with varying heights above sea level. In this exercise, we need to determine the altitude of different locations to see how it affects air density.

To find the altitude of Denver, known as the "mile-high city," we use a conversion. Since 1 foot is approximately 0.305 meters, and there are 5280 feet in a mile, we calculate the altitude by multiplying:

• 5280 feet
• 0.305 meters per foot

Thus, we find that Denver sits approximately 1609.44 meters above sea level. This conversion is crucial when examining physical changes in air properties like density as it varies with altitude.

Quadratic Functions

Quadratic functions play a critical role in this exercise, especially when calculating density changes through the air at various altitudes. The function provided,
\[d(x) = (3.32 \times 10^{-9}) x^2 - (1.14 \times 10^{-4}) x + 1.22\],
illustrates a relationship where density is not linear, but follows a curve based on the quadratic formula. The function captures the change in density (\(d(x)\)) with the variable altitude (\(x\)).

This equation can be solved using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\], where the parameters are:

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

Solving this puts us in position to comprehend altitudes where certain conditions—like where the density exceeds specific values—are satisfied.

Density of Air at Sea Level

The density of air at sea level provides a baseline comparison for understanding how air thins as altitude increases. At sea level, we substitute \(x = 0\) into the function:
\[d(0) = (3.32 \times 10^{-9}) \times 0^2 - (1.14 \times 10^{-4}) \times 0 + 1.22\]
resulting in \(1.22 \, \text{kg/m}^3\).

This density value serves as a critical point for comparison. When we calculate the air density at different locations, like in Denver, it becomes clear why altitudes make environments feel different. Lower air density at higher elevations affects everything from breathing in humans to flight in aircraft, making understanding these numbers practically significant.