Problem 44 - Elementary Algebra

Question

An examination of the winning speeds in the Indianapolis 500 automobile race from 1961 to 1970 produces the equation \(y=1.93 x+137.60\), where \(x\) is the number of years from 1960 and \(y\) is the winning speed. Statistical methods were used to obtain the equation, and, for a given year, the equation gives only the approximate winning speed. Use the equation \(y=1.93 x+137.60\) to find the approximate winning speed in a. 1965 b. 1970 c. 1986

Step-by-Step Solution

Answer

Question: Using the linear equation y = 1.93x + 137.60, where x is the number of years from 1960, find the approximate winning speed for the Indianapolis 500 automobile race in the years 1965, 1970, and 1986. Answer: The approximate winning speeds for the given years are 147.25 mph in 1965, 156.90 mph in 1970, and 187.78 mph in 1986.

Step 1: Determine the value of x for each given year

We are given that x is the number of years from 1960. So, we have to find the difference between the year to calculate and 1960. Let's calculate x for each given year: 1. 1965 => x = 1965 - 1960 = 5 2. 1970 => x = 1970 - 1960 = 10 3. 1986 => x = 1986 - 1960 = 26

Step 2: Calculate the approximate winning speed for 1965

Plug x = 5 into the equation y = 1.93x + 137.60: y = 1.93(5) + 137.60 = 9.65 + 137.60 = 147.25 The approximate winning speed in 1965 is 147.25 mph.

Step 3: Calculate the approximate winning speed for 1970

Plug x = 10 into the equation y = 1.93x + 137.60: y = 1.93(10) + 137.60 = 19.30 + 137.60 = 156.90 The approximate winning speed in 1970 is 156.90 mph.

Step 4: Calculate the approximate winning speed for 1986

Plug x = 26 into the equation y = 1.93x + 137.60: y = 1.93(26) + 137.60 = 50.18 + 137.60 = 187.78 The approximate winning speed in 1986 is 187.78 mph.

Key Concepts

Algebraic Problem-Solving

Understanding algebraic problem-solving is essential to interpret and use mathematical formulas effectively. In the context of the Indianapolis 500 automobile race speed calculation, algebraic problem-solving is showcased when determining the approximate winning speeds for various years. The process involves identifying the variables, locating the correct algebraic equation, and substitively the known values to calculate the desired outcome.

In this instance, the provided equation is of the form \(y = mx + b\), which is the slope-intercept form of a linear equation where \(y\) represents the winning speed, \(m\) is the rate of change (\

the slope), \(x\) is the number of years since 1960, and \(b\) is the initial speed at the year 1960. To solve for the winning speeds, one starts by calculating the value of \(x\) for the given years, which involves simple subtraction to find the number of years since 1960. After that, the value of \(x\) is substituted back into the equation to find \(y\), the approximate winning speed for those years.

This tangible example shows how algebra is useful in making predictions and understanding relationships between variables in real-world scenarios. For students to improve in algebraic problem-solving, it is crucial to practice identifying and manipulating equations and to thoroughly understand the relationships between the different components of an algebraic expression.

Statistical Methods in Algebra

Statistical methods in algebra are used for analyzing data and making predictions. In the provided exercise, these methods have likely been employed to derive the linear equation depicting the relationship between the year and the winning speed of the Indianapolis 500 race. Statistical methods involve collecting data, determining the best fit line—commonly through a process known as linear regression—and then representing this fit with an algebraic equation.

Linear regression is often used to predict the value of a dependent variable based on the value of an independent variable. Here, the dependent variable is the winning speed \(y\), and the independent variable is the number of years since 1960, \(x\). The numbers in the equation, 1.93 and 137.60, signify the slope of the best fit line (rate of speed increase per year) and the y-intercept (initial speed in 1960), respectively.

However, it is essential to remember that the equation provides only an approximation of the winning speed. This is because statistical methods account for variations and uncertainties in observed data, and the output is often a simplified model of reality. By acknowledging the approximations made, one can better understand the applications and limitations of algebra within statistical contexts.

Interpreting Linear Models

Interpreting linear models is a critical skill that involves making sense of the relationships that linear equations represent. With the Indianapolis 500 example, the linear equation \(y = 1.93x + 137.60\) serves as a model that predicts the approximate winning speed of the cars over the years. Interpretation revolves around understanding each component of the equation and how it translates to the reality being described.

The slope of the equation, 1.93, signifies that for every additional year since 1960, the winning speed is expected to increase by approximately 1.93 mph. The y-intercept, 137.60, indicates that the winning speed in the starting year, 1960, would be approximately 137.60 mph. When using this model to predict future or past speeds, such as in 1965, 1970, or 1986, we are extrapolating the trend that was observed in the given timeframe.

It's important to understand that this linear model is based on past data and assumes that the increase in speed remains constant over time. Nonetheless, considering factors such as advancements in technology, regulations, and changes in the race track, the model's predictions might diverge from actual future outcomes. Thus, while linear models are excellent tools for prediction and analysis, they should be applied with an understanding of their limitations and in conjunction with other relevant information.