Answer:
[tex]L(y)=144750y+1998000[/tex]
The slope means that each year the average professional baseball player's salary increased by $144,750 for every year after 2000.
The y-intercept means that in year 0 (2000) the average professional baseball player's salary was $1,998,000
The predicted average salary in 2007 is $3,011,250
(b)
[tex]E(y)=1998000\cdot1.062^y[/tex]The initial value represents the average professional baseball player's salary in year 0 (2000), which was $1,998,000.
The growth factor means that the rate of change increases each year by 1.062 times the previous year's increase.
The predicted average salary in 2007 is $3,044,233.94
Explanation:
The problem gives us two pieces of information:
In year 2000, we call t = 0, the average salary was $1,998,000
In year 2006, we call t = 6, the average salary was $2,866,500
If we want to make a function of the average salary variation over the years, we have two points that must lie in the equation of that function:
(0, 1998000) and (6, 2866500)
For (a) we need to assume that is linear growth. The equation of a line is:
[tex]y=mx+b[/tex]Where:
m is the slope
b is the y-intercept. In this case, since we established the year 2000 as t = 0, b = 1998000
Given two points P and Q, we can find the slope by the formula:
[tex]\begin{gathered} \begin{cases}P=(x_P,y_P){} \\ Q=(x_Q,y_Q)\end{cases} \\ . \\ m=\frac{y_Q-y_P}{x_Q-x_P} \end{gathered}[/tex]Then, if we call:
P = (0, 1998000)
Q = (6, 2866500)
[tex]m=\frac{2866500-1998000}{6-0}=\frac{868500}{6}=144750[/tex]Thus, the equation of the linear growth model is:
[tex]L(t)=144750t+1998000[/tex]Now, we can use this to find a prediction for 2007. 2007 is 7 years since 2000; thus t = 7
[tex]L(7)=144750\cdot7+1998000=1013250+1998000=3011250[/tex]In (b) we assume an exponential growth. The formula for the exponential growth is:
[tex]y=a(1+r)^t[/tex]Where:
a is the initial value. In this case, the average salary in 2000, $1,998,000
r is the ratio of growth. We need to find this value
t is the time in years
Then, we can use the point (6, 2866500), and the fact that a = 1998000:
[tex]2866500=1998000(1+r)^6[/tex]And solve:
[tex]\begin{gathered} \frac{2866500}{199800}=(1+r)^6 \\ . \\ \sqrt[6]{\frac{637}{444}}=1+r \\ . \\ 1.062=1+r \end{gathered}[/tex]We call the term "1 + r" growth factor.
Now, we can write the formula:
[tex]E(t)=1998000\cdot1.062^t[/tex]To find a prediction of the average salary in 2007, we use the function and t = 7:
[tex]E(7)=1998000\cdot1.062^7=1998000\cdot1.5236=3044233.937[/tex]