Problem 85 - Brief Applied Calculus

Question

BEHAVIORAL SCIENCES: Smoking and Educatior According to a study, the probability that a smoker will quit smoking increases with the smoker's educational level. The probability (expressed as a percent) that a smoker with \(x\) years of education will quit is approximately \(y=0.831 x^{2}-18.1 x+137.3\) (for \(10 \leq x \leq 16\) ). a. Graph this curve on the window \([10,16]\) by \([0,100]\). b. Find the probability that a high school graduate smoker \((x=12)\) will quit. c. Find the probability that a college graduate smoker \((x=16)\) will quit.

Step-by-Step Solution

Answer

The probability that a high school graduate will quit is 39.76%, and for a college graduate, it is 60.44%.

Step 1: Understanding the Function

The problem provides a quadratic function: \(y = 0.831x^2 - 18.1x + 137.3\). This function models the probability (in percentage) that a smoker with \(x\) years of education will quit smoking. The domain for \(x\) is \([10, 16]\).

Step 2: Graphing the Function

To graph the function \(y = 0.831x^2 - 18.1x + 137.3\) on the window \([10, 16]\) for \(x\) and \([0, 100]\) for \(y\), plot points for different \(x\) values within the domain. Use a graphing tool or graphing calculator to visualize the curve.

Step 3: Calculating Probability for High School Graduate

Substitute \(x = 12\) into the function to find the probability for a high school graduate. \[y = 0.831(12)^2 - 18.1(12) + 137.3\] Calculate \(y\) to find the probability.

Step 4: Calculate Probability

Perform the arithmetic: \(12^2 = 144\), so \(0.831 \times 144 = 119.664\). Therefore, \[y = 119.664 - 217.2 + 137.3 = 39.764\]. Thus, the probability that a high school graduate smoker will quit is approximately \(39.76\%\).

Step 5: Calculating Probability for College Graduate

Substitute \(x = 16\) into the function to find the probability for a college graduate. \[y = 0.831(16)^2 - 18.1(16) + 137.3\] Calculate \(y\) to find the probability.

Step 6: Calculate Probability

Perform the arithmetic: \(16^2 = 256\), so \(0.831 \times 256 = 212.736\). Therefore, \[y = 212.736 - 289.6 + 137.3 = 60.436\]. Thus, the probability that a college graduate smoker will quit is approximately \(60.44\%\).

Key Concepts

Probability Modeling

Probability modeling is a way of predicting the likelihood of various outcomes. It is particularly useful when studying behaviors or events that do not follow a simple pattern.
In this scenario, probability modeling helps us understand how education level influences the likelihood of quitting smoking. The quadratic function provided: \(y = 0.831x^2 - 18.1x + 137.3\)
is an example of such modeling, where \(x\) represents years of education, and \(y\) represents the probability that a smoker will quit.

• The quadratic nature of the function indicates that the relationship is not linear, meaning that small changes in education level could significantly affect the probability of quitting.
• Such modeling allows researchers to assess trends and make informed predictions.
• The result given by this quadratic model is always a percentage, fitting the definition of probability, which ranges from 0% (impossible) to 100% (certain).

Understanding how to work with these models can give us important insights into behavior patterns. By using probability modeling, we can analyze real-world problems in a mathematical framework, making it easier to draw conclusions from data.

Quadratic Functions in Applied Problems

Quadratic functions often appear in real-life situations, such as physics, economics, and, as in this case, behavioral sciences.
They can model relationships where the effect of the independent variable (like education level) accelerates or decelerates. This can be seen in the equation:\[y = 0.831x^2 - 18.1x + 137.3\]Break it down step-by-step to understand its implications:

• \(x^2\) term: Indicates that changes become more pronounced with higher values of \(x\). This quadratic component reflects accelerated growth in probability as education increases.
• \(x\) term: Shows a linear decrease. Here, the impact of each additional year of education on quitting smoking is generally decreasing.
• Constant term: This fixed value sets the starting point for the model, providing a base value from which changes occur.

By substituting particular \(x\) values, we calculated the probabilities for different education levels:

• Substituting \(x=12\) yields approximately 39.76%, representing high school graduates.
• Substituting \(x=16\) gives about 60.44%, for college graduates.

Quadratic functions like these give insights about thresholds where significant changes begin to occur, making them indispensable tools in applied problems.

Graphing and Interpreting Data

Graphing offers a powerful way to visualize relationships described by functions like our quadratic equation, \(y = 0.831x^2 - 18.1x + 137.3\).
Visual representation helps illustrate abstract concepts, making data more tangible:

• First, determine the graph's domain and range. In this case, \(x\) ranges from 10 to 16, while \(y\) is limited by 0% to 100%.
• Use plotting tools to input the equation and generate a curve.
• You'll see that the curve illustrates how probability rises with more years of education.

The graph helps simplify calculations by visual inspection:

• You can quickly see where peaks and troughs occur.
• It visually highlights trends.
• This approach notes subtle features like whether the effect plateaus or if tipping points exist.

Through graphing and interpretation, complex data becomes relatable and easier to communicate with others. Utilizing such visual tools supports better decision-making and allows engaging with data effectively.