91Ó°ÊÓ

A de Havilland Beaver is a small plane that is capable of an airspeed of about \(250 \mathrm{km} / \mathrm{h}\) in still air. Consider a situation where this plane is flying \(500 \mathrm{km}\) from Lake Athabasca, Saskatchewan, to Great Slave Lake, Northwest Territories. a) Let \(w\) represent the speed of the wind, in kilometres per hour, where \(w\) is positive for a tailwind and negative for a headwind, and \(t\) represent the time, in hours, it takes to fly. What equation represents \(t\) as a function of \(w ?\) What is the non-permissible value for the function? b) Graph the function for a domain that includes its non-permissible value. c) Explain what the behaviour of the function for various values of \(w\) means in this context, including near its non-permissible value. d) Which part(s) of your graph are actually realistic in this situation? Discuss this with a partner, and explain your thoughts.

Step by step solution

01

- Derive the Equation for Time

To find the time, we use the formula for time, which is distance divided by speed. Here, the total distance is 500 km. The plane's airspeed is 250 km/h. If there is a wind speed represented by \(w\), the effective speed of the plane becomes \(250 + w\). Hence, the equation for \(t\) as a function of \(w\) is: \[ t = \frac{500}{250 + w} \]

02

- Determine Non-Permissible Value

A non-permissible value occurs where the denominator of the function is zero because division by zero is undefined. Set the denominator equal to zero and solve for \(w\): \[ 250 + w = 0 \] \[ w = -250 \] So, the non-permissible value for the function is \(w = -250\).

03

- Graph the Function

Graph the function \(t = \frac{500}{250 + w}\) over a range of \(w\) that includes the non-permissible value (\(w = -250\)). The graph will have a vertical asymptote at \(w = -250\), indicating the function is undefined there. Plot points for different values of \(w\) such as \(-300\), \(-200\), \(0\), and \(100\) to visualize the graph.

04

- Analyze Function Behavior

For different values of \(w\): - When \(w\) is a negative value near \(-250\), the time \(t\) becomes very large, indicating it takes very long to reach the destination due to a strong headwind.- As \(w\) approaches zero, the time \(t\) approaches \(2\) hours, indicating no wind impact.- When \(w\) is positive (tailwind), the time \(t\) decreases, showing that it takes less time to reach the destination with a tailwind.

05

- Identify Realistic Values on the Graph

In a realistic scenario, extreme values of \(w\) such as \(-250\) and very high or very low values of \(w\) are not practical. Realistically, \(w\) would be within a manageable range where wind speeds do not excessively hinder or excessively aid the plane’s travel. Values of \(w\) between \(-100\) and \(100\) km/h might be considered realistic for wind speeds.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distance-Speed-Time Formula

Understanding the relationship between distance, speed, and time is crucial for solving problems involving motion. The primary formula is: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] Using this equation, we can determine how long it takes to travel a certain distance at a given speed. This formula helps us derive the function for the time it takes for the plane to travel 500 km at an airspeed of 250 km/h, influenced by wind speed \(w\). Thus, the equation we use becomes: \[ t = \frac{500}{250 + w} \]. This equation tells us how the flight time \(t\) changes with different wind speeds \(w\).

Non-Permissible Values

In mathematics, especially with rational functions, non-permissible values are values that make the function undefined. These occur when the denominator of a fraction equals zero, leading to division by zero. For our equation: \[ t = \frac{500}{250 + w} \], we find the non-permissible value by setting the denominator to zero:\[ 250 + w = 0 \] \[ w = -250 \] Thus, when \(w = -250\), the function is undefined because the plane's effective speed is zero; it cannot move, making the time infinite. This non-permissible value is crucial in understanding the limitations of our model.

Graphing Rational Functions

Rational functions like \( t = \frac{500}{250 + w} \) can be visualized graphically to understand their behavior. When graphing, we need to consider:

• The non-permissible value, where our function has a vertical asymptote.
• Positive values (tailwinds) and negative values (headwinds) of \(w\).

Plotting the function over a range that includes \(w = -250\), we see that the function approaches infinity as \(w\) nears -250. This helps illustrate how the flight time changes with different wind conditions. Plot points like \(w = -300\), \(-200\), \(0\), \(100\) to get a clearer picture.

Vertical Asymptote

A vertical asymptote occurs where the function becomes undefined, indicating that the value of the function increases or decreases without bound. For our equation \( t = \frac{500}{250 + w} \), the vertical asymptote is at \(w = -250\). Here's why: As \(w\) approaches -250 from either side, the denominator approaches zero, making the time \(t\) extremely large (positive or negative infinity depending on the direction of approach). This tells us that a flight with a wind speed of \(w = -250\) would take an infinitely long time to cover the distance, which is physically impossible. Understanding vertical asymptotes helps us grasp the behavior of rational functions at critical points.