Question

A baseball pitcher's earned run average (ERA) is \(A(e, i)=9 e / i\), where \(e\) is the number of earned runs given up by the pitcher and \(i\) is the number of innings pitched. Good pitchers have low ERAs. Assume \(e \geq 0\) and \(i>0\) are real numbers. a. The single-season major league record for the lowest ERA was set by Dutch Leonard of the Detroit Tigers in \(1914 .\) During that season, Dutch pitched a total of 224 innings and gave up just 24 earned runs. What was his ERA? b. Determine the ERA of a relief pitcher who gives up 4 earned runs in one- third of an inning. c. Graph the level curve \(A(e, i)=3\) and describe the relationship between \(e\) and \(i\) in this case.

Short answer

Question: Calculate the ERA for Dutch Leonard and a relief pitcher, then describe the relationship between earned runs and innings pitched when the ERA is 3. Answer: Dutch Leonard's ERA is 0.96, an excellent performance. The relief pitcher's ERA is 108, a poor performance. When the ERA is constant at 3, there is a linear relationship between earned runs and innings pitched, represented by the equation \(e = \frac{1}{3}i\), where for every 3 innings pitched, the pitcher gives up 1 earned run.

Step-by-step solution

Identify the earned runs, innings pitched and formula

We are given that Dutch Leonard pitched 224 innings and gave up 24 earned runs. The formula for ERA is given as \(A(e, i) = \frac{9e}{i}\).

Calculate the ERA

Substitute \(e=24\) and \(i=224\) into the formula: \(A(24, 224) = \frac{9(24)}{224}\). Divide 9 * 24 by 224 to find the ERA: \(A(24, 224) = 0.964286\).

Interpret the ERA

Dutch Leonard's ERA, to two decimal places, is 0.96, which is a very low and impressive ERA, indicating that he was an excellent pitcher during that season. #b. Calculating ERA for the relief pitcher#

Identify the earned runs, innings pitched and formula

We are given that the relief pitcher pitched one-third of an inning and gave up 4 earned runs. The formula for ERA is given as \(A(e, i) = \frac{9e}{i}\).

Calculate the ERA

Substitute \(e=4\) and \(i=\frac{1}{3}\) into the formula: \(A(4, \frac{1}{3}) = \frac{9(4)}{\frac{1}{3}}\). Multiply 9 * 4 and then divide by \(\frac{1}{3}\) to find the ERA: \(A(4, \frac{1}{3}) = 108\).

Interpret the ERA

The relief pitcher's ERA is 108, which is a very high ERA, indicating that the relief pitcher performed poorly in that specific game. #c. Graphing the level curve A(e, i)=3#

Solve for e

To graph the level curve \(A(e, i) = 3\), we first solve for e: \(3 = \frac{9e}{i}\). Then, isolate the e value to find the equation of the level curve. To do this, multiply both sides of the equation by i: \(3i = 9e\). Now divide both sides of the equation by 9: \(e = \frac{1}{3}i\).

Describe the relationship

The level curve is represented by the equation \(e = \frac{1}{3}i\). This curve shows a linear relationship between the earned runs (e) and the innings pitched(i), with a positive slope of \(\frac{1}{3}\). When the ERA is constant at a value of 3, for every 3 innings pitched, the pitcher gives up 1 earned run.

Key concepts

Baseball Statistics

Earned Run Average (ERA) is an essential statistic used in baseball to evaluate the performance of a pitcher. It measures the average number of earned runs a pitcher allows per nine innings pitched. This metric helps determine a pitcher's effectiveness in preventing the opposing team from scoring, with lower ERAs indicating better performance.
The ERA calculation involves:

• The number of earned runs (e) a pitcher allows.
• The number of innings (i) they have pitched.

The formula to calculate ERA is given by \(A(e, i) = \frac{9e}{i}\). Using this formula provides a standardized way to compare pitchers across different games and periods, as every pitcher's performance is normalized to a 9-inning scale.Understanding ERA is crucial for analyzing historic achievements in baseball, such as Dutch Leonard's record-setting ERA of 0.96 in 1914. Such impressive statistics highlight a pitcher's ability to consistently limit scoring opportunities for the opposing teams.

Mathematical Modeling

Mathematical modeling plays a vital role in understanding and calculating baseball statistics like ERA. It involves using mathematical equations and relationships to represent real-world scenarios. In the case of ERA, the formula \(A(e, i) = \frac{9e}{i}\) is a model that encapsulates the relationship between earned runs, innings pitched, and a standardized measure of performance.
This model is

• Practical for predicting outcomes.
• Useful for performance comparisons.
• Helps to identify patterns and relationships in data.

For example, in an exercise where a relief pitcher gives up 4 earned runs in one-third of an inning, the calculation involves multiplying the earned runs by 9 and dividing by the innings pitched. This results in an ERA of 108, starkly demonstrating a poor performance relative to standard expectations.Mathematical models like this one provide valuable insights and can be adjusted or scaled as needed to reflect different dimensions of a pitcher's performance.

Linear Relationships

Linear relationships are fundamental in many mathematical models, including the calculation and graphical representation of ERA. When we set a constant ERA, such as \(A(e, i) = 3\), we derive a linear equation: \(e = \frac{1}{3}i\).
The equation reveals several insights:

• It represents a straight-line relationship between earned runs and innings pitched.
• The slope of the line, \(\frac{1}{3}\), indicates the ratio of earned runs to innings.
• This line helps visualize and interpret changes in performance as innings increase.

In the specific case of a level curve set at an ERA of 3, this means for every 3 innings pitched, a pitcher gives up 1 earned run, underlining the consistent behavior described by the linear equation.Understanding these linear relationships helps in graphing performance, projecting future statistics, and comparing different pitchers' capabilities under standardized scoring conditions.