Certainly! Let's break down the steps required to solve the problem and predict the average programmer's salary in 2010.
Step 1: Understand the function
The given function is:
[tex]\[ y = 31.8 \times (1.06)^n \][/tex]
where:
- [tex]\( y \)[/tex] represents the salary in thousands of dollars.
- [tex]\( n \)[/tex] is the number of years since 1990.
Step 2: Define the range of years
We need to analyze the years from 1990 to 2010. So, we will first list these years and then calculate the number of years since 1990 for each of these.
Years:
[tex]\[ \text{1990, 1991, 1992, ..., 2010} \][/tex]
Corresponding [tex]\( n \)[/tex] values (years since 1990):
[tex]\[ n = 0, 1, 2, \ldots, 20 \][/tex]
Step 3: Calculate the salaries for each year
Let's calculate the salary for each year using the given function:
[tex]\[ y = 31.8 \times (1.06)^n \][/tex]
Step 4: Predict the salary for the year 2010
For the year 2010:
- [tex]\( n = 2010 - 1990 = 20 \)[/tex]
Using the given formula:
[tex]\[ y = 31.8 \times (1.06)^{20} \][/tex]
Now, we will calculate this step-by-step:
1. Calculate [tex]\( 1.06^{20} \)[/tex]:
[tex]\[ 1.06^{20} \approx 3.207135472 \][/tex]
2. Multiply by 31.8:
[tex]\[ y = 31.8 \times 3.207135472 \approx 101.303706999 \][/tex]
So, the predicted average programmer's salary in the year 2010 is approximately [tex]$101.30$[/tex] thousand dollars.
Step 5: Graph the function
To graph the function [tex]\( y = 31.8 \times (1.06)^n \)[/tex] from 1990 to 2010, plot the points for each year and connect them to visualize the growth. The x-axis will represent the years, ranging from 1990 to 2010, and the y-axis will represent the salary in thousands of dollars.
Since I cannot draw the graph here, I'll describe it:
- Label the x-axis from 1990 to 2010.
- Label the y-axis with appropriate salary values, starting from 31.8 (for 1990) upward.
- Plot the salary value for each year and connect the points with a smooth curve.
The graph will show an increasing exponential curve since the base of the exponential [tex]\( (1.06) \)[/tex] is greater than 1, indicating growth over time.
Summary
The average programmer's salary in 2010, according to the model provided, is approximately [tex]$101.30$[/tex] thousand dollars.
