Question

the number of residents of a certain state (in thousands) who are enrolled in medicare t years since 2000 can be represented by the equation M=81t+2681

Answer 1

The equation

`M=81t+2681`

Shows the number of residents enrolled in medicare since the year 2000.

"t" indicates the number of years, it is the independent variable of the function, in the graph, it will correspond to the variable on the x-axis.

"M" indicates the number of residents enrolled in medicare, it is the dependent variable of the function, if you were to graph the equation, this variable would correspond to the y-axis.

a)

The year 2000 corresponds to the initial year taken into account, the starting point can be considered as t=0.

The M-intercept is the value of M when t=0, to determine this value replace t=0 into the equation and calculate the corresponding value of M.:

`\begin{gathered} M=81t+2681 \\ M=81\cdot0+2681 \\ M=2681 \end{gathered}`

The coordinates of the M-intercept are (t,M) = (0,2681)

This means that in the year 2000 (t=0), this state had 2681 thousand residents enrolled in Medicare.

b)

The t-intercept is the value of t when M=0. In the context of the exercise, it will be the year when there are no residents enrolled in Medicare. To determine this value you have to zero the equation and solve it for t:

`\begin{gathered} M=81t+2681 \\ 0=81t+2681 \end{gathered}`

-Subtract 2681 from both sides of the equation

`\begin{gathered} 0-2681=81t+2681-2681 \\ -2681=81t \end{gathered}`

-Divide both sides by 81

`\begin{gathered} -(2681)/(81)=(81t)/(81) \\ -33.09=t\to t\approx-33.1 \end{gathered}`

The t-intercept is -33, this value indicates that 33 years before the starting year, the number of residents enrolled in Medicare was zero.

The coordinates of the t-intercept are: (t,M) = (-33.1,0)

To determine what year it was, you have to subtract 33 from the starting year 2000

`2000-33=1967`

So, in the year 1967, this state had zero thousand residents enrolled in Medicare.