Problem 11 - 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 \(P=101.3 e^{-\frac{h}{26,200}}\). Find a formula for the height in terms of the outside air pressure.

Step-by-Step Solution

Answer

The formula for height is \(h = -26,200 \cdot \ln\left(\frac{P}{101.3}\right)\).

Step 1: Start with the given equation

The relationship between pressure and height is given by the equation \(P = 101.3 e^{-\frac{h}{26,200}}\). Our goal is to express \(h\) in terms of \(P\).

Step 2: Isolate the exponential term

To solve for \(h\), we first divide both sides of the equation by 101.3: \(\frac{P}{101.3} = e^{-\frac{h}{26,200}}\).

Step 3: Apply the natural logarithm

Take the natural logarithm of both sides to eliminate the exponential: \(\ln\left(\frac{P}{101.3}\right) = -\frac{h}{26,200}\).

Step 4: Solve for the height \(h\)

Multiply both sides by \(-26,200\) to solve for \(h\): \(h = -26,200 \cdot \ln\left(\frac{P}{101.3}\right)\).

Key Concepts

Altimeter

An altimeter is a crucial device used in aviation to measure the altitude or height of an aircraft relative to sea level. This measurement is vital for ensuring safe navigation and avoiding potential obstacles. Typically, altimeters function by determining the outside air pressure, which decreases as an airplane ascends into higher altitudes.
Here’s a simple breakdown of how altimeters work:

• Inside the altimeter is a sensitive barometer that measures the outside air pressure.
• As the airplane climbs, the air pressure outside decreases. The altimeter detects this change to calculate the altitude.
• The altimeter translates this measurement into a readable altitude display for the pilot.

A model represented by the equation in the exercise, where pressure relates to height, helps pilots to understand the current height based on external conditions. This is crucial for flight safety and precision.

Natural Logarithm

The natural logarithm is a mathematical function that is the inverse of the exponential function. In equations, it is denoted as \ln\ or \ln(x)\. It is particularly useful for solving equations involving exponential expressions.When the equation in the exercise involves an exponential relationship between air pressure and altitude, using the natural logarithm becomes key to isolating and solving for variables. Here's how it helps:

• By applying the natural logarithm to both sides of an exponential equation, we can eliminate the exponential component.
• For example, in the equation \(e^{-\frac{h}{26,200}}\), applying \ln\ allows us to drop the base \(e\) and isolate the exponent.
• This step is crucial for rearranging the formula to solve for the desired variable \(h\), or height.

The natural logarithm thus simplifies complex exponential relationships, allowing easier manipulation and understanding of equations.

Air Pressure

Air pressure is the force exerted by the weight of air molecules and is a critical factor in aviation. As altitude increases, air pressure decreases. This relationship is fundamental for operations in aviation, for both navigation and understanding weather patterns.In the formula provided in the exercise, the air pressure \(P\) determines the height \(h\) through an exponential relationship:* At sea level, air pressure is at its maximum; as altitude increases, the weight of air above decreases, reducing the pressure significantly.* For an aircraft, knowing the air pressure allows pilots to calculate and monitor altitude via their altimeter.Understanding the connection between air pressure and altitude helps to ensure safety in aviation. This is why equations like the one in the exercise are essential for translating real-world conditions into precise navigational data.