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Chapter 3: Problem 30

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.

Short Answer

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

Step by step solution

01

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.
02

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: \( 3,390,000 = 29,303(1 + r)^{43} \).
03

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.
04

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 \( A \approx 5,751,000 \).
05

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 \( A \approx 8,458,000 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.

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Most popular questions from this chapter

The number of women earning a bachelor's degree from a 4 -yr college in the United States grew from 48,869 in 1930 to approximately 920,000 in 2010. (Source: National Center for Education Statistics.) Find an exponential function that fits the data, and the exponential growth rate, rounded to the nearest hundredth of a percent.

Use a graphing calculator (or Graphicus) to graph each function and find all relative extrema. $$ f(x)=x^{2} e^{-x} $$

The initial weight of a prisoner of war is \(140 \mathrm{lb}\). To protest the conditions of her imprisonment, she begins a fast. Her weight \(t\) days after her last meal is approximated by $$W=140 e^{-0.009 t}$$ a) How much does the prisoner weigh after 25 days? b) At what rate is the prisoner's weight changing after 25 days?

Differentiate. $$ y=\frac{e^{3 t}-e^{7 t}}{e^{4 t}} $$

Spread by skin-to-skin contact or via shared towels or clothing, methicillin- resistant Staphylococcus aureus (MRSA) can easily infect growing numbers of students at a university. Left unchecked, the number of cases of MRSA on a university campus weeks after the first 9 cases occur can be modeled by $$ N(t)=\frac{568.803}{1+62.200 e^{-0.092 t}} $$ a) Find the number of infected students beyond the first 9 cases after 3 weeks, 40 weeks, and 80 weeks. b) Find the rate at which the disease is spreading after 20 weeks. c) Explain why an unrestricted growth model is inappropriate but a logistic equation is appropriate for this situation. Then use a calculator to graph the equation.
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