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Chapter 2: Problem 21

According to the World Wildlife Fund, a group in the forefront of the fight against illegal ivory trade, the price of ivory (in dollars/kilo) compiled from a variety of legal and black market sources is approximated by the function $$ f(t)=\left\\{\begin{array}{ll} 8.37 t+7.44 & \text { if } 0 \leq t \leq 8 \\ 2.84 t+51.68 & \text { if } 8

Short Answer

Expert verified
The graph of the function \(f(t)\) is composed of two line segments, with the first segment going from (0, 7.44) to (8, 74.12) and the second segment going from (8, 74.36) to (30,137.28). At the beginning of 1970 (t=0), the price of ivory was 7.44 dollars/kilo, and at the beginning of 1990 (t=20), the price of ivory was approximately 109.28 dollars/kilo.

Step by step solution

01

(Step 1: Identify the line segment equations)

For the first part of our piecewise function, we have: 1. \( f(t) = 8.37t + 7.44 \) if \( 0 \leq t \leq 8 \) For the second part, we have: 2. \(f(t) = 2.84t + 51.68 \) if \( 8 < t \leq 30 \)
02

(Step 2: Sketch the graph of the function f.)

To sketch the graph of the function, we need to draw both line segments separately. 1. For the first segment, we can find the coordinates for two points on the line by plugging in t=0 and t=8: a. At t=0, \( f(t) = 8.37(0) + 7.44 = 7.44 \). So the first point is (0, 7.44). b. At t=8, \( f(t) = 8.37(8) + 7.44 \approx 74.12 \). So the second point is (8, 74.12). Draw the line segment between these two points. 2. For the second segment, we can find the coordinates for two points on the line by plugging in t=8 and t=30: a. At t=8, \( f(t) = 2.84(8) + 51.68 \approx 74.36 \). So the first point is (8, 74.36). b. At t=30, \( f(t) = 2.84(30) + 51.68 \approx 137.28 \). So the second point is (30, 137.28). Draw the line segment between these two points. The x-axis represents the years (t) and the y-axis represents the price of ivory (f(t)).
03

(Step 3: Find the price of ivory at the beginning of 1970 and 1990.)

To find the price of ivory at the beginning of 1970 (t=0), we will use the equation for the first part of the function: 1. At t=0, 1970: \( f(t) = 8.37(0) + 7.44 = 7.44 \) dollars/kilo. To find the price of ivory at the beginning of 1990 (t=20), we will use the equation for the second part of the function: 2. At t=20, 1990: \( f(t) = 2.84(20) + 51.68 \) \( \approx 109.28 \) dollars/kilo.

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Key Concepts

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

Graph Sketching
Graph sketching is a fundamental skill in mathematics that involves drawing the graph of a function based on its equation. In this case, we sketch the graph of a piecewise function. A piecewise function is a function composed of multiple sub-functions, each with its own domain. To sketch the graph:
  • Identify each segment of the piecewise function. Notice the different equations for specified intervals of the independent variable, here represented by time, \( t \).
  • For each segment, calculate the value of the function at the endpoints of its domain. For instance, for \( f(t) = 8.37t + 7.44 \), we calculate \( f(0) = 7.44 \) and \( f(8) = 74.12 \).
  • Plot the points calculated for each segment and connect them to form line segments. Ensure each line accurately represents its equation and range.
  • Verify any potential points of intersection between segments as these signify changes in the equation.
By following these steps, the graph of a piecewise function can be effectively depicted. Understanding how each component works together is crucial for accurate graph sketching.
Function Analysis
Function analysis involves studying the behavior and properties of a mathematical function. Analyzing a piecewise function requires understanding each sub-function separately and together.
  • Determine the domain of each piece: The domain is where the function is defined. For \( f(t) = 8.37t + 7.44 \), the domain is \( 0 \leq t \leq 8 \); for \( f(t) = 2.84t + 51.68 \), it's \( 8 < t \leq 30 \).
  • Observe continuity and points of transition: Each piece should smoothly transition or connect from one segment to the next. Check the function's value and slope at these intersections.
  • Identify key characteristics: For instance, determining values at specific points such as \( t=0 \) or \( t=20 \), helps in understanding how the function behaves over time.
  • Consider the slopes of the lines: The slope tells how rapidly the price changes over time. A steeper slope in the initial years (like \( 8.37 \)) means a rapid increase compared to a gentler slope later (\( 2.84 \)).
This analysis helps in gaining a deeper insight into the nature and behavior of the function and the real-world phenomenon it models.
Mathematical Modeling
Mathematical modeling is the process of representing real-world situations through mathematical equations and functions. In this exercise, the price of ivory over time is modeled using a piecewise function.
  • Set up the problem: Understand the context, which here involves the fluctuating price of ivory due to market dynamics.
  • Choose a suitable function: For complicated scenarios, using a piecewise function, allows different behaviors to be expressed within different intervals. The model chosen reflects this by splitting the time into intervals with different trends.
  • Interpret the model: Analyze the equations. The significant increase in price from \( 1970 \) to \( 1978 \) caught by a steeper line indicates harsher market conditions. A slower increase after indicates changing circumstances such as regulations affecting the market.
  • Validate the model: Compare predictions with actual data if available. This ensures the model's reliability in explaining and predicting future trends.
Through careful mathematical modeling, complex real-world behavior can be captured and understood, aiding in decision-making and fostering insight. Here, it reveals trends in the ivory market's pricing over two decades.

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