Problem 59 - Applied Calculus

Question

Read the Application Preview on pages \(3-4\) a. Use the regression line \(y=-0.356 x+257.44\) to predict the world record in the year \(2020 .\) [Hint: If \(x\) represents years after \(1900,\) what value of \(x\) corresponds to the year 2020 ? The resulting \(y\) will be in seconds, and should be converted to minutes and seconds.] b. According to this formula, when will the record be 3 minutes 30 seconds? [Hint: Set the formula equal to 210 seconds and solve. What year corresponds to this \(x\) -value?

Step-by-Step Solution

Answer

a. 3 minutes 35 seconds b. Year 2034

Step 1: Identify the value of x for the year 2020

To predict the world record for the year 2020 using the regression equation \(y = -0.356x + 257.44\), we first need to determine the value of \(x\). Since \(x\) represents years after 1900, for the year 2020, \(x = 2020 - 1900 = 120\).

Step 2: Calculate y using the regression equation

With \(x = 120\), plug this value into the regression equation: \[y = -0.356(120) + 257.44.\]Calculating this gives:\[y = -42.72 + 257.44 = 214.72.\]Thus, the predicted world record time in seconds is 214.72.

Step 3: Convert time from seconds to minutes and seconds

Since \(214.72\) seconds need to be expressed in minutes and seconds, divide 214 by 60 to get:\[ ext{Minutes} = rac{214}{60} = 3.5783 ext{ minutes}.\]The integer part (3) is the number of minutes. The fractional part (0.5783) is converted back to seconds by multiplying by 60:\[ ext{Seconds} = 0.5783 imes 60 = 34.70.\]Therefore, the prediction for 2020 is 3 minutes and approximately 35 seconds.

Step 4: Set up the equation for 3 minutes 30 seconds

To find when the world record will be exactly 3 minutes and 30 seconds, first convert this time to seconds: \[3 imes 60 + 30 = 210 ext{ seconds}.\]Now, set the regression equation equal to 210:\[210 = -0.356x + 257.44.\]

Step 5: Solve for x

Subtract 257.44 from both sides of the equation:\[210 - 257.44 = -0.356x.\]Simplifying gives:\[-47.44 = -0.356x.\]Divide both sides by -0.356:\[x = rac{-47.44}{-0.356} \ x = 133.25.\]

Step 6: Identify the corresponding year

Since \(x\) is 133.25 (years after 1900), add this to 1900 to find the year:\[1900 + 133.25 = 2033.25.\]Since a time cannot occur in a fraction of a year, round up to the next year, 2034.

Key Concepts

Understanding Regression Analysis

Regression analysis is a powerful statistical tool used to explore the relationship between variables. In this context, it helps predict the behavior of one variable based on the value of another. For example, in our exercise, we are using a linear regression line equation \( y = -0.356x + 257.44 \) to determine world record times depending on the year.

• Independent Variable: In our example, this is the year, more precisely, the number of years after 1900.
• Dependent Variable: This is the expected world record time in seconds.

By inputting different values of \( x \), we can predict the future or reconstruct the past records. Regression models like this allow us to spot trends and make educated guesses about future data points.

Introduction to Predictive Modeling

Predictive modeling involves using statistical techniques to forecast outcomes. In our case, we use the provided regression equation to predict athletic world record times.When predicting, it's crucial to:

• Define the Model: Understand the equation used for prediction. Here, it's a simple linear model where the line's slope and intercept give insights into future predictions.
• Input the Variables: Use the correct value of \( x \) to ensure that predictions are accurate. In this exercise, \( x = 120 \) for the year 2020.

Predictive modeling uses such models to:

• Anticipate future outcomes.
• Assist decision-making processes.
• Improve strategic planning and allocation of resources.

Mathematical Problem Solving Techniques

Solving mathematical problems often involves breaking down the problem into simpler, more manageable steps. Let's apply this approach to our exercise.First, we translate the real-world scenario into a mathematical one. Using the given regression model, we are solving for different instances like predicting the year for a specific record time.

• Step-by-step Calculation: Follow each step logically, as seen in the example where we calculated \( y \) for the year 2020 and reversed the process for a specific time target.
• Conversion and Interpretation: After solving for \( y \), or \( x \), convert results into comprehensible formats—seconds into minutes, or fractions into whole numbers (years).

This problem-solving approach is not only useful for academic exercises but also in practical scenarios where precise solutions are necessary. Understanding the sequence and logic behind each step is fundamental for resolving any mathematical challenge.