Problem 36 - Applied Calculus

Question

Soybean production, in millions of tons $$\begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 2000 & 2001 & 2002 & 2003 & 2004 & 2005 \\ \hline \text { Production } & 161.0 & 170.3 & 180.2 & 190.7 & 201.8 & 213.5 \\ \hline \end{array}$$ 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. $$\begin{array}{c|c|c|c|c|c} \hline \text { Elevation (ft) } & \text { Sca level } & 1000 & 2000 & 3000 & 4000 \\ \hline \text { Takeoff roll (ft) } & 670 & 734 & 805 & 882 & 967 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The takeoff roll as an exponential function of elevation is given by \( y = 670 \times e^{0.000089x} \).

Step 1: Understand the Data

We have two sets of data given in the exercise. The first set involves soybean production over several years, but this data is not directly needed for solving the problem related to the takeoff roll. The relevant data involves how takeoff roll changes concerning elevation for an aircraft.

Step 2: Identify the Problem

Our task is to find a formula for the takeoff roll as an exponential function of the airport elevation using the given data points: Sca level (670 ft roll), 1000 ft (734 ft roll), 2000 ft (805 ft roll), 3000 ft (882 ft roll), and 4000 ft (967 ft roll).

Step 3: Formulate the Exponential Model

The general exponential model can be written as: \[ y = a imes e^{bx} \] where \( y \) is the takeoff roll, \( x \) is the elevation, \( a \) is the initial value (roll at sea level), and \( b \) is the exponential growth rate. We have the initial value \( a = 670 \) (the roll at sea level).

Step 4: Use Known Data Points to Determine Parameters

We need to calculate the value of \( b \), the growth rate. To find \( b \), use another data point. Let's use the takeoff roll value for 1000 ft:\[ 734 = 670 imes e^{1000b} \]Divide both sides by 670:\[ e^{1000b} = \frac{734}{670} \]Take the natural log of both sides:\[ 1000b = \ln\left(\frac{734}{670}\right) \]Solve for \( b \):\[ b = \frac{\ln\left(\frac{734}{670}\right)}{1000} \]

Step 5: Calculate the Exponential Growth Rate

Using the formula from Step 4, compute:\[ b = \frac{\ln\left(\frac{734}{670}\right)}{1000} \approx 0.000089 \]Thus, our complete model is:\[ y = 670 imes e^{0.000089x} \]

Step 6: Validate the Model

Use another data point to check if the model is accurate. For example, test with 2000 ft:\[ y = 670 imes e^{0.000089 imes 2000} \]\[ y \approx 805 \]Since this aligns closely with the given data (805 ft at 2000 ft elevation), the model is valid.

Key Concepts

Aircraft Takeoff Roll

When an airplane is taking off, it needs a specific distance on the runway to reach the necessary speed to lift off the ground. This distance is known as the takeoff roll.

Takeoff rolls are longer at higher altitudes because the air is less dense. This lower air density means the aircraft must travel faster to achieve the required lift, thus requiring a longer runway.

The given dataset illustrates how different airport elevations impact the takeoff roll for an aircraft. The data points show that as the elevation increases, the takeoff roll distance increases as well. This relationship can be modeled using an exponential function, which we'll explore further in this article.

Airport Elevation

Airport elevation refers to the height above sea level at which the airport is located. It is an essential factor influencing flight operations, especially during takeoff and landing.

Due to air density decreasing with increasing elevation, it affects how aircraft perform. Higher elevation requires more runway for takeoff because the engines and wings have to work harder in less dense air to achieve the necessary lift.

The dataset provides takeoff roll lengths at various elevations - sea level, 1000 ft, 2000 ft, 3000 ft, and 4000 ft. This range helps in understanding how exponentially increasing runway distances are required at elevated airports.

Exponential Growth Model

In mathematics, an exponential growth model is used to describe processes that increase proportionally to their current value, often as a function of time or, in this case, elevation.

The general form of an exponential function is: \[ y = a \times e^{bx} \] where \( y \) represents the outcome, \( a \) is the initial value, \( e \) is the base of natural logarithms, \( b \) is the growth rate, and \( x \) is the independent variable (elevation in this scenario).

In the context of the aircraft takeoff roll, the initial value \( a \) is the takeoff distance at sea level, and the parameter \( b \) is calculated from the data to model how the takeoff roll increases exponentially with airport elevation. This model allows us to predict takeoff distances at elevations not specifically measured in the dataset.

Natural Logarithm Calculation

The natural logarithm is a mathematical function that helps in transforming exponential growth models into a linear form, making it easier to solve for the growth rate.

In this context, to determine the growth rate \( b \) in the exponential growth model for takeoff roll, we need to solve the transformation: \[ 1000b = \ln\left(\frac{734}{670}\right) \] Taking the natural logarithm transforms the equation into a linear relationship, allowing us to solve for \( b \) by simplifying and rearranging the equation.

Once \( b \) is found, it completes the formula \( y = 670 \times e^{0.000089x} \) by inserting the calculated value. This functionality is crucial in building an accurate model that reflects the relationship between airport elevation and takeoff roll distance.