Problem 30 - Calculus and its applications

Question

In \(1970,\) the average salary of Major League baseball players was \(\$ 29,303 .\) In \(2013,\) the average salary was \(\$ 3,390,000 .\) Assuming exponential growth occurred, what was the growth rate to the nearest hundredth of a percent? What will the average salary be in \(2020 ?\) In 2025 ? Round your answers to the nearest thousand.

Step-by-Step Solution

Answer

The growth rate is 7.89%. The salary in 2020 is approximately $5,751,000 and in 2025 is $8,458,000.

Step 1: Understand exponential growth formula

Exponential growth can be represented by the formula \( A = P(1 + r)^t \), where \( A \) is the amount after time \( t \), \( P \) is the initial amount, \( r \) is the growth rate, and \( t \) is the time elapsed.

Step 2: Set up the equation for the growth rate

Given: \( P = 29,303 \), \( A = 3,390,000 \), and \( t = 2013 - 1970 = 43 \) years. Substitute these values into the formula: \( 3,390,000 = 29,303(1 + r)^{43} \).

Step 3: Solve for the growth rate

Divide both sides by \( 29,303 \): \( \frac{3,390,000}{29,303} = (1 + r)^{43} \). Compute the left side: 115.7048. Take the 43rd root: \( 1 + r = (115.7048)^{1/43} \). Compute \( r = (115.7048)^{1/43} - 1 \), resulting in \( r \approx 0.0789 \) or 7.89% per year.

Step 4: Calculate salary for 2020 using exponential formula

Use \( r = 0.0789 \), \( P = 29,303 \), and \( t = 2020 - 1970 = 50 \). Find \( A \): \( A = 29,303(1 + 0.0789)^{50} \). Calculate to get \( A \approx 5,751,000 \).

Step 5: Calculate salary for 2025 using exponential formula

Use \( r = 0.0789 \), \( P = 29,303 \), and \( t = 2025 - 1970 = 55 \). Find \( A \): \( A = 29,303(1 + 0.0789)^{55} \). Calculate to get \( A \approx 8,458,000 \).

Key Concepts

Exponential Growth Formula

Exponential growth is a powerful concept that describes how quantities increase rapidly over time. This kind of growth is depicted by the formula \( A = P(1 + r)^t \). In this equation:

• \( A \) represents the amount after a certain time \( t \)
• \( P \) stands for the initial amount or principal
• \( r \) is the growth rate expressed as a decimal
• \( t \) is the duration over which growth is observed

To apply this formula effectively, it's crucial to identify each variable. In the context of Major League baseball salaries, \( P \) was $29,303 in 1970, and \( t \) was the number of years between the two salary figures under consideration. Understanding this setup is essential before calculating the growth rate and projections.

Growth Rate Calculation

Calculating the growth rate is about discovering how rapidly a quantity like a salary increases over time. To find this rate, we rearrange the exponential growth formula to solve for \( r \).First, plug in the known values into the formula: \( 3,390,000 = 29,303(1 + r)^{43} \). By dividing both sides by 29,303, you simplify the equation to \( (1 + r)^{43} = 115.7048 \).Next, you need to take the 43rd root of 115.7048 to isolate \( 1 + r \). This results in \( 1 + r \approx 1.0789 \). Finally, subtract 1 to solve for \( r \):\[ r \approx 0.0789 \times 100 = 7.89\% \]This tells us the average annual growth rate was about 7.89% between 1970 and 2013. It's this growth rate that then allows us to project future salaries.

Future Value Projection

Predicting future values using exponential growth requires you to project forward using the growth rate found. Here’s how this applies to future salary projections:For 2020, substitute the known values into the exponential formula: \( A = 29,303(1 + 0.0789)^{50} \). Calculating this gives an approximate salary of \(5,751,000.To extend this projection to 2025, adjust \( t \) to 55 years since 1970: \( A = 29,303(1 + 0.0789)^{55} \). This results in an estimated salary around \)8,458,000.

• This method allows economists, analysts, and interested observers to make educated guesses about financial progression.
• It highlights how exponential growth can lead to surprisingly high future values.

Understanding these calculations helps in realizing the power of steady growth over time.