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Chapter 3: Problem 12

According to the Statistical Abstract of the United States (www.census.gov), there were approximately 31, 000 crimes reported in the United States in 1998 , and this was dropping by a rate of about 2900 per year. a) On a sheet of graph paper, make a coordinate system and plot the 1998 data as a point. Note that you will only need to graph the first quadrant of a coordinate system, since there are no data for years before 1998 and there cannot be a negative number of crimes reported. Use the given rate to find points for 1999 through 2006 , and then draw a line through your data. We are constructing a continuous model for our discrete situation. b) The following table lists the number of crimes reported, C, the year, and the number of years since 1998. In 1998 , the number was 31,000 and no occurrences of the 2900 decrease had happened yet, so we fill in \(31000-2900(0)\). In 1999 , one occurrence of the 2900 decrease had happened because one year had passed since 1998 . Therefore, the number of crimes reported was \(31000-2900(1)\). And the pattern continues in this manner. Fill in the remaining entries. c) Observing the pattern in the table, we come up with the equation \(\mathrm{C}=31000-2900 \mathrm{x}\) to relate the number of crimes \(\mathrm{C}\) to the number of years \(\mathrm{x}\) after 1998 . Here, \(\mathrm{C}\) is a function of \(\mathrm{x}\), and so we can use the notation \(\mathrm{C}(\mathrm{x})=31000-2900 \mathrm{x}\) to emphasize this. i. Compute C(5). ii. In a complete sentence, explain what C(5) represents. iii. Compute C(8). iv. In a complete sentence, explain what \(\mathrm{C}(8)\) represents.

Short Answer

Expert verified
C(5) = 16500, representing crimes in 2003; C(8) = 7900, representing crimes in 2006.

Step by step solution

01

Plotting the Initial Data Point

To start, we form a coordinate system focusing on the first quadrant since we don't have data for years before 1998, and negative crime numbers are impossible. For 1998, where x = 0, plot the point (0, 31000) for 1998.
02

Calculating Points for Subsequent Years

Using the rate of 2900 crimes decrease per year, calculate the number of crimes for the years 1999 through 2006. Each point (x, C) will follow the formula C = 31000 - 2900x. For instance, for x = 1 (1999), C = 31000 - 2900 * 1 = 28100.
03

Filling in the Table

Fill in the table using the formula C = 31000 - 2900x for each year between 1999 and 2006. For example, for 2000 (x = 2), calculate C = 31000 - 2900 * 2 = 25200, and continue this calculation for each subsequent year.
04

Creating the Equation of the Model

The pattern observed gives rise to the equation C(x) = 31000 - 2900x, where C is a function of x, the number of years since 1998. This linear function will model the decline in incidents.
05

Computing C for Specific Years

First, compute C(5) using C(x) = 31000 - 2900x. Plugging x=5 gives C(5) = 31000 - 2900*5 = 16500. Next, compute C(8) in the same way: C(8) = 31000 - 2900*8 = 7900.
06

Interpreting the Calculated Values

C(5) represents the number of crimes reported in the year 2003, which is 16500 crimes. C(8) represents the number of crimes reported in the year 2006, which is 7900 crimes.

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Key Concepts

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

Functions
In mathematics, functions play a crucial role in expressing relationships between variables. A function is essentially a rule that assigns a unique output to each given input. In this exercise, we establish a function to represent the yearly decline in reported crimes starting from 1998. The function is written as \( C(x) = 31000 - 2900x \), where \( C \) represents the number of crimes, and \( x \) is the number of years since 1998. This function is a linear equation, which means it forms a straight line when graphed. The negative coefficient of \( x \), \(-2900\), indicates a decrease in crime rates each year. Understanding functions helps us to predict future outcomes and analyze patterns in data.
Coordinate System
Coordinate systems are a foundational component in graphing mathematical functions. They allow us to visually represent data and functions on a plane. In this exercise, we use a Cartesian coordinate system, which consists of two perpendicular axes: the horizontal (x-axis) and vertical (y-axis). Each point on the plane is represented by a pair of numerical coordinates. Since we are looking at years from 1998 onward and crime should be positive, our graph uses only the first quadrant.
  • The x-axis represents years since 1998.
  • The y-axis represents the number of crimes reported.

By plotting points such as (0, 31000) for 1998, (1, 28100) for 1999, and so on, we can draw a line to illustrate how crime figures decrease yearly according to our function.
Data Interpretation
Data interpretation involves analyzing and making sense of collected data to draw meaningful conclusions. In the exercise, we analyze the table of crime statistics over several years using our derived function \( C(x) = 31000 - 2900x \).
  • For 1998: \( x = 0 \), so \( C(0) = 31000 \)
  • For 1999: \( x = 1 \), so \( C(1) = 28100 \)
  • For subsequent years, we continue to calculate using the function.

The values computed help in understanding trends and patterns, such as predicting crime numbers in future years and offering insight into the effectiveness of crime prevention strategies.
Statistical Analysis
Statistical analysis is about collecting and exploring data to identify patterns and trends. In this problem, statistical analysis enables us to use past crime data to predict future values. By utilizing our linear equation \( C(x) = 31000 - 2900x \), a key aspect is understanding how the statistical abstract's data rebuffs or supports predictions about crime trends.
  • 1998 has 31,000 reported cases, setting the primary data point.
  • Annual decrement of 2,900 cases is a stable, predictable pattern.

By computing specific years like \( C(5) \) and \( C(8) \), which stand for 2003 and 2006 respectively, we delve into the effectiveness of strategies implemented, offering data-driven insights. Statistical analysis transforms raw figures into narratives about societal progress and challenges.

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Most popular questions from this chapter

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