Problem 69 - Brief Applied Calculus

Question

Smoking and Income Based on a recent study, the probability that someone is a smoker decreases with the person's income. If someone's family income is \(x\) thousand dollars, then the probability (expressed as a percentage) that the person smokes is approximately \(y=-0.31 x+40\) (for \(10 \leq x \leq 100)\) a. Graph this line on the window \([0,100]\) by \([0,50]\). b. What is the probability that a person with a family income of $$\$ 40,000$$ is a smoker? [Hint: Since \(x\) is in thousands of dollars, what \(x\) -value corresponds to $$\$ 40,000 ?]$$ c. What is the probability that a person with a family income of $$\$ 70,000$$ is a smoker? Round your answers to the nearest percent.

Step-by-Step Solution

Answer

Probability for $40,000 income: 28%. Probability for $70,000 income: 18%.

Step 1: Understanding the Equation

The given equation is \(y = -0.31x + 40\), where \(x\) is the income in thousands of dollars, and \(y\) is the probability in percentage that a person smokes. The range for \(x\) is from 10 to 100.

Step 2: Graph the Equation

To graph the equation \(y = -0.31x + 40\), plot it on the coordinate plane. The x-axis represents income in thousands \([0, 100]\), and the y-axis represents the smoking probability in percentage \([0, 50]\). The line will have an intercept at \((0, 40)\) and a slope of \(-0.31\), meaning it will decrease as \(x\) increases.

Step 3: Calculate Probability for $40,000 Income

A \$40,000 income corresponds to \(x = 40\) (since \(x\) is in thousands). Substitute \(x = 40\) into the equation: \(y = -0.31(40) + 40\). Calculate \(-0.31 \times 40 = -12.4\), so \(y = 40 - 12.4 = 27.6\). Rounded to the nearest percent, the probability is 28%.

Step 4: Calculate Probability for $70,000 Income

A \$70,000 income corresponds to \(x = 70\). Substitute \(x = 70\) into the equation: \(y = -0.31(70) + 40\). Calculate \(-0.31 \times 70 = -21.7\), so \(y = 40 - 21.7 = 18.3\). Rounded to the nearest percent, the probability is 18%.

Key Concepts

Probability

Probability is a way to measure the likelihood of an event happening. In this context, we are looking at the probability that someone is a smoker based on their family income. The concept of probability helps us quantify uncertainty, with values ranging from 0% (impossible) to 100% (certain). In applied calculus, probability can be linked to continuous variables, like income, and adapted into linear equations.

This specific scenario uses a linear function to determine smoking probability. The given formula is expressed as a linear equation:

• y = -0.31x + 40, where x is the income in thousands, and y is the probability percentage.

This formula implies that as income (x) increases, the probability (y) decreases. Understanding how to interpret probability in real-life situations, like income affecting smoking habits, helps in making informed decisions based on data. With applied calculus, these linear relationships provide a clear understanding of dynamic changes in probability.

Linear Equations

Linear equations are mathematical expressions that describe a straight-line relationship between two variables. They have the general form y = mx + b, where m is the slope and b is the y-intercept. In this problem, the equation y = -0.31x + 40 depicts how smoking probability (y) varies with income (x).

The slope -0.31 tells us how much y decreases for a unit increase in x. Here, every increase of $1000 in income results in a decrease of 0.31% in the likelihood of smoking. The y-intercept 40 indicates the starting probability when income is zero.

Recognizing the role of both slope and intercept is key to analyzing how variables relate through linear equations. It’s a foundational concept in applied calculus that aids in understanding complex relationships, such as economic behavior patterns or trends in population health.

Graphing Functions

Graphing is a visual way to interpret and analyze the relationship between variables expressed through functions. In this example, graphing the equation y = -0.31x + 40 provides a visual insight into how income influences smoking probability.

To start graphing, identify key points such as the y-intercept (0, 40)—where the line crosses the y-axis—and the slope, which determines the line’s steepness. We can then plot this line over the interval [0,100] for income on the x-axis and [0,50] for probability on the y-axis. As x increases, the graph descends, showing a declining trend in smoking likelihood as income rises.

Graphing functions in applied calculus helps visually summarize data and interpret mathematical relationships. When faced with real-world problems like this exercise, graphing can make understanding trends and patterns much more intuitive and accessible.