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Chapter 5: Problem 15

In southwest Georgia, the average pine pulpwood prices vary predictably over the course of the year, primarily because of weather. From 1993 through 1997, prices followed a similar pattern. In the first quarter of each year, the average price \(P\) was \(\$ 18.50\) per ton. It decreased at a steady rate to \(\$ 14\) in the second quarter and then increased at a steady rate up to \(\$ 18\) by the fourth quarter. \({ }^{51}\) a. Sketch a graph of pulpwood prices as a function of the quarter in the year. b. What type of function is \(P\) from the first to the second quarter? c. What formula for price \(P\) as a function of \(t\), the quarter, describes the price from the first to the second quarter? d. What type of function is \(P\) from the second to the fourth quarter? e. What formula for price \(P\) as a function of \(t\), the quarter, describes the price from the second to the fourth quarter? f. Write a formula for price \(P\) throughout the year as a piecewise-defined function of \(t\), the quarter.

Short Answer

Expert verified
P is linear from Q1 to Q2 and Q2 to Q4, described by piecewise function: \[ P(t) = \begin{cases} -4.5t + 23, & 1 \leq t < 2 \\ 2t + 10, & 2 \leq t \leq 4 \end{cases} \]

Step by step solution

01

Graph the Price Function

The quarters are plotted on the x-axis ranging from 1 to 4, and the price on the y-axis. For the first quarter, the price is 18.50, it decreases to 14 for the second quarter, and then increases again, reaching 18 in the fourth quarter. The graph will show a linear decline from the first to the second quarter and a linear increase from the second through the fourth quarter, resembling a V shape.
02

Identify Function Type from Q1 to Q2

From Q1 to Q2, the price decreases steadily from 18.50 to 14. This indicates a linear relationship where the price is decreasing over time.
03

Formula for Price from Q1 to Q2

Since it's linear, use point-slope form to find the formula. The change in price is \( 14 - 18.50 = -4.50 \) over one quarter, giving a slope \( m = -4.50 \). Hence, the formula is \( P(t) = -4.50t + 23 \) for quarters \( t \) in [1, 2].
04

Identify Function Type from Q2 to Q4

From Q2 to Q4, the price increases steadily from 14 to 18. This is a linear relationship where the price increases over time.
05

Formula for Price from Q2 to Q4

With a steady increase from Q2 to Q4, the slope is \( \frac{18 - 14}{4 - 2} = 2 \). Therefore, the linear equation would be starting from Q2: \( P(t) = 2(t - 2) + 14 \), which simplifies to \( P(t) = 2t + 10 \) for quarters \( t \) in [2, 4].
06

Complete Piecewise Function for Q1 to Q4

Combine the formulas from the previous steps into a piecewise function. \[ P(t) = \begin{cases} -4.5t + 23, & 1 \leq t < 2 \ 2t + 10, & 2 \leq t \leq 4 \end{cases} \] This formula defines the price of pulpwood for each quarter based on the year.

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Key Concepts

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

Linear Equations
A linear equation is a mathematical statement that describes a straight line on a graph. It has the general form:
  • \( y = mx + b \)
Here, \( m \) represents the slope of the line, which indicates the rate of change, while \( b \) signifies the y-intercept, where the line crosses the y-axis.
In the problem, the linear equations are used to model the changes in prices of pine pulpwood over specific quarters. From Q1 to Q2, the price changes described are linear, meaning they decrease evenly by a set amount each quarter. This predictability in the rate of change is a key feature of linear equations.
A clear understanding of linear equations helps in modeling real-world scenarios, like predicting future prices or costs based on past behavior patterns. Understanding the components of a linear equation allows one to interpret the data and anticipate trends effectively.
Graphing Functions
Graphing functions involves plotting points on a coordinate plane to depict the relationships described by equations. Each point corresponds to a particular input and output, shown as \( (x, y) \). This visual representation makes it easier to understand and interpret data trends.
In this exercise, graphing was essential to portray the trend of pine pulpwood prices over different quarters. With the quarters on the x-axis and price on the y-axis, one can clearly visualize the function's behavior across the year.
The graph takes a V shape, indicating both a decline and a rise in prices over certain periods. This visual cue assists in identifying where the function changes direction, allowing a better grasp of even complex relationships.
By understanding how to graph functions, one can quickly decipher how different variables interact, making it a crucial skill in analyzing and interpreting data efficiently.
Function Modeling
Function modeling involves creating equations or functions to describe real-world situations accurately. This process helps in predicting future behaviors based on historical data. In this example, the prices of pine pulpwood across quarters are modeled using piecewise linear functions.
To adequately represent the changes in prices, two distinct linear functions model two separate phases of the year. The first models the decrease from Q1 to Q2, and the second models the increase from Q2 to Q4.
Function modeling provides a framework for making informed predictions, aligning with specific patterns observed over time. By breaking down the cycle into manageable segments, complex patterns become simpler and easier to analyze. Understanding how to create and interpret these models is essential in applying mathematical concepts to real situations.
Rate of Change
The rate of change is a measure of how much a quantity changes concerning another, often tied to time. It is essentially the slope of the line in a linear function and is a depiction of how rapidly or slowly the values are changing.
In linear equations, this is represented by the slope \( m \). In the context of this problem, the rate of change shows how rapidly the price of pine pulpwood decreases from Q1 to Q2 and how it increases from Q2 to Q4.
  • From Q1 to Q2, the rate of change is \(-4.50\), indicating a steady decrease.
  • From Q2 to Q4, the rate is \(2\), pointing to a gradual increase.
By understanding the rate of change, one can infer the speed and direction of trends, which is invaluable in planning and forecasting in various domains.

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Most popular questions from this chapter

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