Problem 42 - 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. $$\begin{array}{c|c|c|c|c|c} \hline \text { Elevation (ft) } & \text { Sea level } & 1000 & 2000 & 3000 & 4000 \\ \hline \text { Takeoff roll (ft) } & 670 & 734 & 805 & 882 & 967 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The exponential function is \( y = 670 \cdot 1.000092^x \).

Step 1: Identify the independent and dependent variables

In the given exercise, the airport elevation (in feet) is the independent variable, and the takeoff roll (in feet) is the dependent variable. We are finding an exponential function that describes the relationship between these two variables.

Step 2: Set up the exponential model

The general form of an exponential function is \( y = ab^x \) where \( y \) is the takeoff roll, \( x \) is the elevation, \( a \) is the initial value (when \( x = 0 \)), and \( b \) is the base or growth factor. From the data, we know \( y = 670 \) when \( x = 0 \), indicating \( a = 670 \). The goal is to determine the value of \( b \).

Step 3: Calculate the growth factor

Select two points from the table to calculate \( b \). Using the points (0, 670) and (1000, 734), substitute into the formula:\[734 = 670b^{1000}.\]Solve for \( b \) by dividing both sides by 670:\[b^{1000} = \frac{734}{670}.\]Take the 1000th root to solve for \( b \):\[b = \left(\frac{734}{670}\right)^{1/1000} \approx 1.000092.\]

Step 4: Construct the exponential function

Now substitute the values of \( a \) and \( b \) into the general form to get the equation:\[y = 670 \cdot 1.000092^x.\]This function gives the takeoff roll as an exponential function of the airport elevation.

Step 5: Verify the model

Plug in another data point from the table (e.g., when elevation = 2000) into the exponential function to check its accuracy:\[y = 670 \cdot 1.000092^{2000} \approx 805.\]This matches the given takeoff roll for 2000 ft elevation, confirming that the model is correct.

Key Concepts

Independent and Dependent Variables

In the world of mathematics and science, understanding the relationship between different quantities often involves identifying independent and dependent variables. In the context of the exercise given, the airport elevation is the independent variable, while the takeoff roll for the aircraft is the dependent variable. This means that changes in the airport elevation trigger changes in the required takeoff roll distance.

• The **independent variable**, elevation, can be thought of as the input in this scenario. You can adjust it freely, without being influenced by any other factor in the problem.
• The **dependent variable**, which is the takeoff roll distance, relies on the elevation. As you increase or decrease the elevation, the dependent variable's value is directly impacted by these changes.

The concept of independent and dependent variables is fundamental because it helps set the stage for building a mathematical model, such as an exponential function, which describes how these variables interact.

Exponential Growth

Exponential growth is a specific way in which a quantity increases over time. In cases of exponential relationships, the rate of growth is proportional to the current quantity. Imagine a snowball rolling down a hill, growing larger as it gathers more snow with each roll. Similarly, in our exercise, the takeoff roll of the aircraft grows as the airport elevation increases.The general form of an exponential function is given by the formula:\[ y = ab^x \]Where:

• \( y \) is the output value or the dependent variable, representing the takeoff roll here.
• \( a \) is the initial amount or starting value when the independent variable \( x \) is 0; in this case, it's the takeoff roll at sea level.
• \( b \) is the base or growth factor illustrating the rate at which the variable grows.

In the exercise, we determined the base to be approximately 1.000092, signifying a slow yet steady increase in the takeoff roll as the elevation heightens. This steady increase showcases the nature of exponential growth, as opposed to linear growth that increases by a constant amount.

Mathematical Model

A mathematical model is an abstract representation using mathematical language to describe a system or a phenomenon in the real world. In this exercise, we aim to represent how takeoff roll distances increase with airport elevation. This model comes alive through the exponential function we've constructed.To build our model, we:

• Identified our variables (elevation and takeoff roll) and chose an appropriate mathematical form (exponential function).
• Determined key constants needed for our model: the initial value of \( a = 670 \) at sea level and the growth factor \( b \approx 1.000092 \).

The resulting function:\[ y = 670 \cdot 1.000092^x \]This formula not only represents the relationship between elevation and takeoff roll but also allows predictions for other elevations. By substituting different elevation values into the model, one can forecast how long the takeoff roll will be at any given height, enabling pilots to plan accordingly.Mathematical models are invaluable across various fields, providing insights and predictive power in everything from engineering to economics.