Problem 123 - College Algebra with Modeling and Visualization

Question

Runvay Length There is a relation between an airplane's weight \(x\) and the runway length \(L\) required for takeoff. For some airplanes the minimum runway length \(L\) in thousands of feet may be modeled by \(L(x)=3 \log x,\) where \(x\) is measured in thousands of pounds. (Sourcet. I. Haefner, Introduction to Thangortation Systems.) (a) Graph \(L\) for \(0

Step-by-Step Solution

Answer

(a) Graph: logslope; (b) Runway doesn't increase 10x; (c) Logarithms increase by plus constant.

Step 1: Understanding the Function

The function given is the minimum runway length in thousands of feet, expressed as \( L(x) = 3 \log x \), where \( x \) is the airplane's weight in thousands of pounds. This means if an airplane weighs \( 10,000 \) pounds, \( x = 10 \), and if \( 50,000 \) pounds, \( x = 50 \). We need to graph this for \( 0 < x \leq 50 \).

Step 2: Graphing L(x) for the Range

To graph \( L(x) = 3 \log x \), calculate some key points in the domain \( 0 < x \leq 50 \). For example, at \( x = 1 \), \( L(1) = 3 \log 1 = 0 \). At \( x = 10 \), \( L(10) = 3 \log 10 = 3 \). At \( x = 50 \), \( L(50) = 3 \log 50 \approx 8.5 \). Plot these points and sketch a smooth curve through them. The graph is a curve that increases slowly and logarithmically.

Step 3: Interpreting the Graph

The graph shows a logarithmic growth, which means initially as the weight increases, the runway length increases significantly, but as the weight becomes much larger, the inclusion rate of the runway length grows more slowly. This gives insight into how runway length requirements scale with airplane weight.

Step 4: Evaluating Tenfold Increase

Calculate \( L(x) \) for \( 10,000 \) pounds (\( x = 10 \)): \( L(10) = 3 \log 10 = 3 \). \( L(x) \) for \( 100,000 \) pounds (\( x = 100 \)): \( L(100) = 3 \log 100 = 6 \). The required runway length increases from 3 to 6, doubling, not tenfold.

Step 5: Generalization of the Tenfold Increase

If an airplane's weight increases tenfold, from \( x \) to \( 10x \), the function calculations show \( L(10x) = 3 \log(10x) = 3 (\log x + \log 10) = 3 \log x + 3 \). Thus, the runway length increases by an additive constant (\(3\) in this case), not multiplicative. Therefore, the increase isn't by a factor of 10.

Key Concepts

Graph Interpretation

In the exercise, you're tasked with graphing and interpreting the function \(L(x) = 3 \log x\), which models the minimum runway length needed for an airplane's takeoff based on its weight. The core idea here is understanding how the logarithmic graph behaves. Logarithmic functions, like this one, grow quickly at first and then slow down as the input values increase. This means:

• At lower weights, the runway length increases sharply.
• As the weight becomes very large, the increase in runway length slows down.

Visualizing the graph allows you to capture this behavior: if \(x\) represents the weight and \(L(x)\) the required runway, the curve will start at \(L(1) = 0\), rise quickly, and then gradually flatten out. Graphing it provides an intuitive sense of how additional weight impacts runway needs.

Mathematical Modeling

Mathematical modeling helps us translate real-world scenarios into mathematical expressions. In our scenario, the function \(L(x) = 3 \log x\) serves as a model that predicts the runway length needed for different airplane weights. The logarithmic function here reflects real-world dynamics:

• Each point on the function represents the minimum runway length for a specific airplane weight.
• Logarithmic models account for diminishing returns; adding more weight requires proportionately less runway increase compared to earlier additions in weight.

This type of modeling is critical in transportation planning, where resources such as runway length need to be efficiently allocated based on estimated needs. By using \(\log\)-based models, calculations take into account practical limits on infrastructure expansion.

Understanding Function Behavior

Understanding function behavior involves seeing how changes in inputs affect the outputs in a function. For \(L(x) = 3 \log x\), interpreting how the runway length changes with airplane weight helps us explore function properties like scaling and growth rates. When the weight increases tenfold, such as from \(10,000\) to \(100,000\) pounds:

• Calculate the function at \(x = 10\): \(L(10) = 3 \log(10) = 3\).
• Calculate at \(x = 100\): \(L(100) = 3 \log(100) = 6\).

The results show a doubling of runway length, not a tenfold increase. This illustrates a key point about logarithmic functions: they increase additively when the input is multiplied, rather than multiplicatively. Understanding this can help when predicting how changes in one aspect of a problem influence other connected attributes.