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Chapter 1: Problem 60

For each \(\$ 1\) increase in the price of a \(\$ 300\) plane ticket. an airline will lose 60 passengers, so if the ticket price is increased to \(\$ x,\) the decrease in passengers is modeled by \(60(300-x)\)

Short Answer

Expert verified
The decrease in passengers for each $1 increase in ticket price is calculated by function \(60*(300-x)\).

Step by step solution

01

Understanding the problem

The problem provides a function \(60(300-x)\) which models the decrease in number of passengers for every $1 increase in the plane ticket price. To understand the behavior of this function, we can simply substitute various values.
02

Substituting values into the function

We start by substituting an initial value for x. Let's take x equal to $301. Thus, the decrease in passengers when price increases by $1 to $301 is \(60*(300-301)= -60\). The negative value implies a decrease in passengers by 60.
03

Continue substituting values

Let's substitute some more values for x into the function to trace the pattern. For x = $302, the decrease in passengers is \(60*(300-302)= -120\). Similarly, for x = $303, the decrease is \(60*(300-303)= -180\). The pattern continues as such where for each $1 increase, there is a decrease of 60 passengers.
04

Conclusion

It is clear from the steps above that the function \(60*(300-x)\) correctly models the relationship of number of passengers decreasing with increase in ticket price. For every $1 increase in ticket price, the airline loses 60 passengers.

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Key Concepts

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

Mathematical Modeling
Mathematical modeling is an essential tool in analyzing and solving real-world problems. In our daily lives, we often encounter situations where we need to predict how a change in one variable might influence another. In the exercise we're exploring, we deal with a real-world scenario of how ticket prices influence the number of passengers.Here, mathematical modeling allows us to convert this scenario into a mathematical equation: \(60(300-x)\). This model accurately predicts the decrease in passengers based on the ticket price adjustments. By setting up and using this mathematical function, we can:
  • Understand the direct relationship between ticket price and passenger numbers.
  • Predict the outcomes of changing ticket prices with ease.
  • Help airlines make informed decisions based on financial scenarios.
By using these models, airlines can optimize their pricing strategy to find the balance between maximizing passengers and maintaining a profitable ticket price.
Functions in Algebra
In algebra, functions are a fundamental concept used to describe mathematical relationships. A function essentially tells us how one variable depends on another. In the given problem, the function \(60(300-x)\) represents how the number of passengers is affected by the price increase of tickets.This specific function is a linear function. Linear functions in algebra are characterized by the fact that they show a constant rate of change. They are usually expressed in the form \(y = mx + b\), where \(m\) is the slope and \(b\) the y-intercept.In our scenario:
  • The slope is \(-60\), representing the loss of 60 passengers per dollar increase in ticket price.
  • The expression \((300-x)\) inside the function indicates the tickets price change from the initial rate.
The beauty of linear functions is in their simplicity and predictability. They allow us to quickly evaluate how changes to one variable (in this case the ticket price \(x\)) impact another (passenger count).
Problem Solving in Mathematics
Problem solving in mathematics involves identifying, analyzing, and resolving mathematical issues through structured approaches. In this scenario, we are tasked with understanding how passenger numbers change with ticket pricing, a classic example of applying mathematical solutions to practical problems.Breaking Down the Steps:The step-by-step solution includes understanding the problem, substituting values, and identifying a pattern. Here's how we can approach such problems effectively:
  • Understand the Problem: Start by reading the problem carefully to identify variables and the relationships between them.
  • Apply the Model: Use the given function, \(60(300-x)\), to substitute values (e.g., \(x = 301, 302, \) etc). Notice how the function's output reflects the changes in passenger numbers.
  • Identify Patterns: As you substitute different values, observe the consistent pattern of passenger reduction per ticket price increment.
  • Draw Conclusions: By recognizing the pattern, conclude the impact on passengers for each price increase. This confirms the function's validity and assists in making decisions.
Utilizing these steps ensures clarity and precision in solving mathematical problems. By practicing these techniques, students can develop a systematic approach toward tackling similar mathematical challenges.

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Most popular questions from this chapter

a. Use a graphing utility to graph \(f(x)=x^{2}+1\) b. Graph \(f(x)=x^{2}+1, g(x)=f\left(\frac{1}{2} x\right),\) and \(h(x)=f\left(\frac{1}{4} x\right)\) in the same viewing rectangle. c. Describe the relationship among the graphs of \(f, g,\) and \(h,\) with emphasis on different values of \(x\) for points on all three graphs that give the same \(y\) -coordinate. d. Generalize by describing the relationship between the graph of \(f\) and the graph of \(g,\) where \(g(x)=f(c x)\) for \(0

Sketch the graph of \(f\) using the following properties. (More than one correct graph is possible.) \(f\) is a piecewise function that is decreasing on \((-\infty, 2), f(2)=0, f\) is increasing on \((2, \infty),\) and the range of \(f\) is \([0, \infty)\)

As in Exercise \(22,\) you have 600 feet of fencing to enclose a rectangular field. However, one side of the field lies along a canal and requires no fencing. Express the area of the field, \(A,\) as a function of one of its dimensions, \(x .\)

Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. $$(4,7) \text { and }(8,10)$$

The regular price of a computer is \(x\) dollars. Let \(f(x)=x-400\) and \(g(x)=0.75 x\) a. Describe what the functions \(f\) and \(g\) model in terms of the price of the computer. b. Find \((f \circ g)(x)\) and describe what this models in terms of the price of the computer. c. Repeat part (b) for \((g \circ f)(x)\) d. Which composite function models the greater discount on the computer, \(f^{\circ}\) g or \(g \circ f\) ? Explain.
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