Question

Gary just started a new job as a nurse. he is given a starting salary of $58,550 per year. he is also told that his salary will increase to $71,950 at the beginning of 10 years. what is the rate of change in gary's salary?

Answer 1

Firts we need to find the rate of change, or in other words, the slope of the line.

Question 1:

a)

For this we can take two points in the form (years, salary). Then we can define as year 0 the year when Gary starts to work. In this year the salary is $58,550. The first point is (0, $58,550)

The next point we can take is at the year 10, when the salary of Gary will be $71,950. The second point is (10, $71,950)

b) Now that we have the two points, we can use the slope formula to get the rate of change. The slope formula is, for two points A and B:

$\begin{gathered} \begin{cases}A(x_a,y_a) \\ B(x_b,y_b)\end{cases} \\ m=\frac{y_a-y_b}{x_a-x_b} \end{gathered}$

In this case, we can call the points A(0, $58,550) and B(10, $71,950). Using the formula:

$m=\frac{58,550-71,950}{0-10}=\frac{-13400}{-10}=1340$

c) "The rate of change in Gary's salary is $1340 per year."

Question 2:

a) The slope intercept form of a line is:

$y-y_1=m(x_{}-x_1)$

Where:

y is the output of the function.

x is the input of the function. (we provide the function with a value for x and the function give us a value of y)

m is the slope of the line. We calculate it in question 1.

x1 is the x coordinate of a point we choose.

y1 is the y-coordinate of the same point of x1.

In this case, we know:

m = 1340;

And we can take the point (0, $58,550), thus:

x1 = 0

y1 = 58,550

b) Now we need to use all this values and use the slope-intercept form:

$y-58,550=1340(x-0)$

And solve to get:

$y=1340x+58,550$

Question 3:

a) Now we have a equation for the salary, we can use this to find the salary in 13 years. We just need to replace x = 13 in the equation:

$\begin{gathered} \begin{cases}y=1340x+58,550 \\ x=13\end{cases} \\ y=1340\cdot13+58,550 \\ y=75,970 \end{gathered}$

B) The salary of Gary in 13 years will be $75,970.