Question

The average professional baseball player's salary \(S\) (in millions of dollars) from 1995 to 2006 can be modeled by \(S=0.1527 t+0.294, \quad 5 \leq t \leq 16\) where \(t\) represents the year, with \(t=5\) corresponding to 1995 (see figure). Use the model to predict the year in which the average professional baseball player's salary exceeds \(\$ 3,000,000\). (Source: Major League Baseball)

Short answer

Thus, according to the model, the average professional baseball player's salary will exceed $3,000,000 in the year 2013.

Step-by-step solution

Adjust the given value

The given value that we must exceed is $3,000,000. However, the function provided expresses \(S\) in terms of millions of dollars. Therefore, we must convert this value into millions, which gives us $3 million.

Set the function greater than 3

Now we will substitute $3 million for \(S\) in our function. This will allow us to solve for \(t\). \[0.1527 t + 0.294 > 3\] Next, we’ll isolate \(t\) by first subtracting 0.294 on both sides of the inequality, this gives us \[0.1527 t > 2.706\] Then, we'll divide the resulting inequality by the coefficient of \(t\), which is 0.1527, on both sides, this gives us \(t > 17.71692\)

Interpret the result

The obtained value is \(t > 17.71692\). This indicates that the salary will exceed \$3 million just after the 17th year from 1995. The next whole year would be the 18th year, which, in this context, represents the calendar year 2013. Because year values should be integers, we would round the result to the next greatest integer.

Key concepts

Understanding Linear Equations

Linear equations are the bedrock of algebra and serve as an essential tool for modeling various real-world scenarios, such as predicting salaries. A linear equation can usually be written in the form of y = mx + b, where y is the dependent variable, m is the slope of the line representing the rate of change, x is the independent variable, and b is the y-intercept representing the starting value.

In the context of the baseball player's salary problem, we have the linear equation S = 0.1527t + 0.294. Here, S represents the salary in millions of dollars, t represents the years since 1995, 0.1527 is the annual increase in salary, and 0.294 reflects the salary at the starting point (1995). This linear model helps us extrapolate information to predict future salaries or analyze past trends.

By understanding the structure of the linear equation, we can interpret the coefficients and their impact on the salary progression over time, which is fundamental when working on predictions and analyses within the field of mathematics or economics.

Solving Inequalities in Algebra

Inequalities, like equations, are statements about the relative size of two expressions, with symbols such as <, <=, >, >= instead of an equals sign. Algebraic inequalities are powerful tools for representing conditions and constraints in a model.

When we adjusted the given salary value from dollars to millions and replaced the salary S in the function with \(3 million, we got the inequality 0.1527t + 0.294 > 3. To find when the salary exceeds \)3 million, we solve this inequality for t. The solution process involves isolating the variable on one side, just like in equations, but with extra caution: when we multiply or divide by a negative number, the direction of the inequality sign must be reversed. However, in our case, the coefficient of t is positive, so the inequality's direction is unchanged.

Solving inequalities demands precision and understanding of the rules governing these mathematical statements. It provides us the ability to answer not just specific questions (like with equations) but also to explore a range of possible solutions within a given constraint.

Modeling with Algebra

Modeling with algebra is the application of algebraic techniques to represent and solve problems from real-life situations. It's a vital skill that helps us connect mathematical theories with everyday experiences.

Our exercise involves creating a model — in this case, a linear equation — to predict future salaries for baseball players. The equation S = 0.1527t + 0.294 converts abstract mathematical concepts into a tangible tool for making predictions. When modeling, we start by identifying the variables and constants that will form our algebraic expressions. The rate of increase 0.1527t in our model shows the continuous annual growth in salary, while the constant 0.294 represents the initial salary value.

Additionally, determining the range of our model is also critical. The inequality 5 <= t <= 16 sets the boundaries for our prediction (from 1995 to 2006). This restrains our model to a specific period and provides accuracy to our prediction. Furthermore, we use algebraic operations and understanding of inequalities to determine when the salary will exceed a specific amount. Through mathematical modeling, complex situations are simplified, enabling us to analyze and make sense of the world around us.