Problem 53 - Algebra 1

Question

The average salary for a professional baseball player in the United States can be approximated by \(y=283(1.2)^{t},\) where \(t=0\) represents the year \(1984 .\) Using this approximation, find the ratio of an average salary in 1988 to an average salary in 1994

Step-by-Step Solution

Answer

The ratio of the average salary in 1988 to the average salary in 1994 is \(\frac {283(1.2)^4}{283(1.2)^{10}}.\)

Step 1: Define the Years

Firstly,Subtract 1984 from both 1988 and 1994 to find how many years past 1984 these years are. So for 1988, \(t=1988-1984=4\). And for 1994, \(t=1994-1984=10\).

Step 2: Find the Salaries

Use these values for \(t\) to find the average salary for these years. By substituting \(t=4\) into the formula, the average salary in 1988 is \(y_1 =283(1.2)^4\). By substituting \(t=10\) into the formula, the average salary in 1994 is \(y_2 =283(1.2)^{10}\).

Step 3: Calculate the Ratio

Compute the ratio of the average salary from 1988 to the average salary from 1994 by dividing \(y_1\) by \(y_2\). So the final ratio will be \(\frac {y_1}{y_2} = \frac {283(1.2)^4}{283(1.2)^{10}}\).

Key Concepts

Average Salary Calculation

Calculating average salary using an exponential formula involves understanding how salaries change over time. In this exercise, the salary calculation is modeled by an exponential function, which is common in financial mathematics. The formula used here is:
\[ y = 283(1.2)^t \]
This equation means that the average salary (\( y \)) starts at 283 units in 1984 when \( t = 0 \) and grows by a rate of 20% each year (since 1.2 represents a 20% growth). To find the salary for a specific year, like 1988 or 1994, you plug in the value of \( t \) representing how many years have passed since 1984.

• For 1988, \( t = 4 \) (1988 minus 1984).
• For 1994, \( t = 10 \) (1994 minus 1984).

Once the values of \( t \) are identified, it's straightforward to substitute these into the formula to find the salary for each year.

Ratio Calculation

After obtaining the salaries for two different years, the next task is to find the ratio of these salaries. A ratio is essentially a comparison of two numbers. In terms of salaries, it's about understanding how much more one amount is compared to another.
In the given problem, the formula for the salary checkpoints is:
\[ y_1 = 283(1.2)^4 \] \[ y_2 = 283(1.2)^{10} \]These represent the salaries in 1988 and 1994, respectively. To calculate the ratio of the salary in 1988 to the salary in 1994, you divide one by the other:
\[ \frac{y_1}{y_2} = \frac{283(1.2)^4}{283(1.2)^{10}} \]
The 283 terms cancel each other out, which simplifies the problem greatly, leaving:
\[ \frac{(1.2)^4}{(1.2)^{10}} \]
This simplifies further to \((1.2)^{-6}\), which is \( \frac{1}{(1.2)^6} \). This calculation shows how much less the average salary was in 1988 compared to 1994 in exponential terms.

Exponential Functions

Exponential functions are prevalent in scenarios involving growth or decay, such as population growth, radioactive decay, and financial investments - like salary growth. An exponential function has the general form:
\[ y = a \, b^{x} \]
Where \( a \) is the starting amount, \( b \) is the growth (or decay) factor, and \( x \) is the exponent representing time or another variable.In this case, the exponential function \( y = 283(1.2)^t \) describes salary growth over time. Here's a breakdown of the parts:

• \( 283 \): Initial salary in the base year (1984).
• \( 1.2 \): Growth factor. Since it's greater than 1, it indicates a percentage increase.
• \( t \): Time in years since 1984.

What makes exponential functions unique is the rate of increase. As \( t \) increases, the salary grows at an ever-accelerating rate, making this function a powerful tool for modeling real-world financial data where growth is based on percentage increases over time.