Problem 12 - Algebra 2

Question

The altimeter in an airplane gives the altitude or height \(h(\text { in feet })\) of a plane above sea level by measuring the outside air pressure \(P\) (in kilopascals). The height and air pressure are related by the model \(P=101.3 e^{-\frac{h}{26,200}}\). Use the formula you found in Exercise 11 to approximate the height of a plane above sea level when the outside air pressure is 57 kilopascals.

Step-by-Step Solution

Answer

The plane is approximately 15,084 feet above sea level.

Step 1: Understand the Given Model

The model given in the problem is \( P = 101.3 e^{-\frac{h}{26,200}} \), where \( P \) is the outside air pressure in kilopascals, and \( h \) is the height in feet above sea level. We need to determine how this model is used to find the height when the air pressure is 57 kilopascals.

Step 2: Plug in the Known Pressure Value

Insert the given air pressure value into the model: \( 57 = 101.3 e^{-\frac{h}{26,200}} \). This equation will allow us to solve for \( h \), the height above sea level.

Step 3: Isolate the Exponential Term

Divide both sides of the equation by 101.3 to isolate the exponential term: \( e^{-\frac{h}{26,200}} = \frac{57}{101.3} \).

Step 4: Solve for the Exponent

Calculate \( \frac{57}{101.3} \) to simplify the right side: \( e^{-\frac{h}{26,200}} \approx 0.5625 \). To solve for \( h \), take the natural logarithm of both sides of the equation: \[ -\frac{h}{26,200} = \ln(0.5625) \].

Step 5: Solve for Height \( h \)

Calculate the natural logarithm: \( \ln(0.5625) \approx -0.5754 \). Substitute this into the equation: \( -\frac{h}{26,200} = -0.5754 \). Multiply both sides by -26,200 to find \( h \): \( h \approx -0.5754 \times -26,200 \).

Step 6: Final Calculation

Perform the multiplication: \( h \approx 15,084 \). This is the approximate height of the plane above sea level when the air pressure is 57 kilopascals.

Key Concepts

Altimeter

An altimeter is a device used in aircraft to measure altitude, which is the height above sea level. It does this by detecting changes in air pressure as the aircraft ascends or descends. The higher the altitude, the lower the air pressure becomes. Altimeters are crucial for pilots to maintain safe flying heights, especially when navigating through varying terrains or avoiding obstacles.

• Altimeters rely on the relationship between air pressure and altitude.
• They are essential for safe aviation operations, helping to maintain proper flight levels.

Much like a compass helps a pilot navigate directionally, an altimeter helps to navigate vertically, ensuring that the aircraft is flying at the intended altitude. With the help of mathematical models, like the equation provided in the exercise, pilots can accurately ascertain their altitude based on the external air pressure.

Air Pressure

Air pressure, often measured in kilopascals (kPa), is the force that air exerts on the surface. As you move higher above sea level, whether by mountain climbing or flying in an airplane, the air pressure decreases. This is because the atmosphere becomes less dense with altitude. Mathematically, this relationship is often represented using exponential functions. In the given model, air pressure decreases exponentially as altitude increases, represented by the formula: \[ P = 101.3 e^{-\frac{h}{26,200}} \]

• Air pressure decreases with increasing altitude.
• Lower air pressure means a higher altitude in the atmospheric model.

Understanding this concept is vital in various fields, not just aviation, but also in weather forecasting, sports, and even health, as changes in air pressure can affect human physiology and the flight of sports equipment, like golf and footballs.

Natural Logarithm

The natural logarithm, abbreviated as "ln," is a mathematical function that is the inverse of the exponential function associated with the constant \(e\). It plays a crucial role in mathematics and applications involving exponential growth or decay, such as the altitude-air pressure relationship in aviation. When solving the given problem, the natural logarithm helps to isolate and solve for the variable \(h\) (height) in the equation. After substituting the known air pressure of 57 kilopascals into the model, the natural logarithm is used to undo the exponential component: \[ -\frac{h}{26,200} = \ln(0.5625) \]

• The natural logarithm helps to simplify and solve exponential equations.
• Essential for reversing exponential growth or decay processes.

By applying the natural logarithm, one can transform the exponential equation into a simpler form that can be solved using basic algebraic operations. This step is vital to determine the height \(h\) from the altitude model provided.