Problem 31 - Applied Calculus

Question

Aircraft require longer takeoff distances, called takeoff rolls, at high altitude airports because of diminished air density. The table shows how the takeoff roll for a certain light airplane depends on the airport elevation. (Takeoff rolls are also strongly influenced by air temperature; the data shown assume a temperature of \(0^{\circ} \mathrm{C}\).) Determine a formula for this particular aircraft that gives the takeoff roll as an exponential function of airport elevation. $$ \begin{array}{l|c|c|c|c|c} \hline \text { Elevation (ft) } & \text { Sea level } & 1000 & 2000 & 3000 & 4000 \\ \hline \text { Takeoff roll (ft) } & 670 & 734 & 805 & 882 & 967 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The takeoff roll is modeled as \( y = 670 \, e^{0.000091x} \).

Step 1: Understand the Problem

We need to determine a formula for the takeoff roll as an exponential function of airport elevation using the given data points. An exponential function typically has the form \( y = ab^x \), where \( y \) is the takeoff roll and \( x \) is the elevation.

Step 2: Identify the Form of the Exponential Function

Identify that the general form of our function is \( y = a \, e^{bx} \), where \( a \) and \( b \) are constants to be determined and \( x \) represents the elevation.

Step 3: Create Equations from Data

We have pairs of values: \( (0, 670) \), \( (1000, 734) \), etc. Substitute these into our equation: \( 670 = a \, e^{b \times 0} \), \( 734 = a \, e^{b \times 1000} \), and continue for other data points.

Step 4: Calculate the Initial Value \( a \)

Using the first pair \( (0, 670) \), solve for \( a \). Since \( e^{b \times 0} = 1 \), then \( a = 670 \). Thus, the function simplifies initially to \( y = 670 \, e^{bx} \).

Step 5: Calculate the Growth Rate \( b \)

Divide the equations for two consecutive data points to eliminate \( a \). For example, \( \frac{734}{670} = e^{1000b} \). Solve \( 1.095522388 = e^{1000b} \) using logarithms: \( b = \frac{\ln(1.095522388)}{1000} \).

Step 6: Solve for \( b \)

By calculating \( b = \frac{\ln(1.095522388)}{1000} \), find \( b \approx 0.000091 \). Use additional data points similarly to verify or refine \( b \).

Step 7: Construct the Exponential Function

With \( a = 670 \) and \( b \approx 0.000091 \), our exponential function becomes \( y = 670 \, e^{0.000091x} \).

Step 8: Verify the Function

Substitute other data points into your equation to check the accuracy of your function. Ensure that the computed values are close to the provided data values.

Key Concepts

Takeoff Roll

The takeoff roll of an aircraft refers to the distance it needs to travel on a runway in order to gain enough speed for takeoff. This crucial aspect is influenced by various factors such as the weight of the aircraft, engine power, and importantly, environmental conditions such as air density. During the takeoff roll:

• An aircraft must reach a specific speed known as the rotation speed to become airborne.
• This speed is directly connected to the lift needed to clear the runway safely and efficiently.

Higher runway distances are essential at locations where certain conditions reduce the aircraft's ability to generate lift quickly.

Airport Elevation

Airport elevation is a key component affecting an aircraft’s takeoff performance. It refers to the height of the airport above sea level. As elevation increases:

• The air becomes less dense, thus reducing the amount of lift generated at a given speed.
• Pilots require longer distances to achieve the necessary lift for safe takeoff.

Thus, airports located at higher elevations, such as those in mountainous areas, present challenges that pilots must carefully plan for. The relationship between airport elevation and takeoff roll can be modeled effectively using exponential functions as seen in our exercise, which helps pilots anticipate and compensate for these differences when planning flights.

Air Density

Air density plays a crucial role in aviation, especially during takeoff. It is defined as the mass of air per unit volume and is affected by factors such as temperature, altitude, and pressure. Key effects of air density on takeoff include:

• Higher air density provides more air molecules for wings to work with, increasing lift at a given speed.
• Conversely, lower air density at higher elevations or temperatures results in less lift, necessitating longer takeoff rolls.

In the context of our problem, air density is constant at a temperature of 0°C, allowing us to isolate and study the effect of elevation alone. Understanding these principles helps in assessing how elevation changes influence an aircraft’s required takeoff roll, ensuring safety and performance are maintained under various environmental conditions.