Question

Although some sports experts debunk this theory, many people believe that the amount of money a baseball team spends on payroll affects the number of wins, and thus its standing in the league. For the 2014 Major League Baseball season, the following linear equation can be used to describe the relationship between an American League team's payroll x (in millions of dollars) and the number of games y that team won during the regular season. y = 0.024x + 79.44. Round each answer to the nearest whole.

a. According to this equation, how many games would have been won in 2014 by a team with a payroll of $90 million?
b. The New York Yankees had a payroll of $204 million in 2014. According to this equation, how many games should they have won during the season?
c. According to this equation, what payroll would have been necessary in 2014 to have won 95 games during the season?
d. Find and interpret the slope of the equation.

Answer 1

a. A team with a payroll of $90 million would have won approximately 82 games in the 2014 season. b. The New York Yankees should have won approximately 84 games in the 2014 season with a payroll of $204 million. c. A payroll of approximately $648 million would have been necessary to win 95 games in the 2014 season. d. The slope of the equation, 0.024, represents the change in the number of games won for every $1 million increase in payroll.

Step-by-step explanation:

a. To find the number of games won by a team with a payroll of $90 million, we substitute x = 90 into the equation y = 0.024x + 79.44. y = 0.024(90) + 79.44 = 2.16 + 79.44 = 81.60. Therefore, a team with a payroll of $90 million would have won approximately 82 games in the 2014 season.

b. To find the number of games the New York Yankees should have won with a payroll of $204 million, we substitute x = 204 into the equation y = 0.024x + 79.44. y = 0.024(204) + 79.44 = 4.896 + 79.44 = 84.336. Therefore, the New York Yankees should have won approximately 84 games in the 2014 season.

c. To find the payroll necessary to have won 95 games in the 2014 season, we substitute y = 95 into the equation y = 0.024x + 79.44. 95 = 0.024x + 79.44. Subtracting 79.44 from both sides, we get 95 - 79.44 = 0.024x. Simplifying, we have 15.56 = 0.024x. Dividing both sides by 0.024, we get x ≈ 648.33. Therefore, a payroll of approximately $648 million would have been necessary to win 95 games in the 2014 season.

d. The slope of the equation, 0.024, represents the change in the number of games won for every $1 million increase in payroll. This means that, on average, for every additional $1 million spent on payroll, a team can expect to win approximately 0.024 more games.