The table list data regarding the average salaries of several
professional athlets in the year 1991 and 2001
Year Average Salary
1991 $252,000
2001 $1,350,000
a) Use the data points to find a linear function that fits the data.
b) Use the function to predict the average salary in 2005 and 2010.
A linear function that fits the data is S(x)=
slope = m = change in salary/change in year = (1350000-252000)/(2001-1991)
so
m = 109,800 dollars/year
so
salary =109,800 (year -1991) + 252,000
that is a linear function of year but if you want it in y = mx + b form then
salary = 109,800*year - 218,359,800
check 2001 back in
salary = 109,800*2001 - 218359800
= $1,350,000 check
To find a linear function that fits the data, we need to determine the relationship between the years and the average salary. In this case, we have two data points (1991, $252,000) and (2001, $1,350,000).
To find the linear function, we can use the slope-intercept form of a linear equation, which is y = mx + b, where y is the dependent variable (average salary), x is the independent variable (year), m is the slope, and b is the y-intercept.
a) Finding the slope (m):
We can calculate the slope using the formula:
m = (y2 - y1) / (x2 - x1)
Plugging in the values from our data points:
m = (1,350,000 - 252,000) / (2001 - 1991)
m = 1,098,000 / 10
m = 109,800
b) Finding the y-intercept (b):
To find the y-intercept, we can substitute the values of one of the points into the equation and solve for b. Let's use the first data point (1991, $252,000):
252,000 = 109,800 * 1991 + b
252,000 = 218,015,800 + b
b = 252,000 - 218,015,800
b = -217,763,800
Now that we have the values of m and b, we can write the linear function:
S(x) = 109,800x - 217,763,800
b) Predicting the average salary in 2005 and 2010:
To predict the average salary in 2005 and 2010, we can substitute the respective years into the linear function we found.
For 2005:
S(2005) = 109,800 * 2005 - 217,763,800
S(2005) = 219,437,000 - 217,763,800
S(2005) ≈ $1,673,200
For 2010:
S(2010) = 109,800 * 2010 - 217,763,800
S(2010) = 220,596,000 - 217,763,800
S(2010) ≈ $2,832,200
Therefore, the predicted average salary in 2005 would be approximately $1,673,200, and in 2010 it would be approximately $2,832,200.