Question

solve the questions below. show your work.

1. the number of unwatched shows in sylvias queue is 85. this number grows by 20 unwatched shows per week.

a. what function represents the relation between the number of unwatched shows, n, and the time, t, measured in weeks?
b. how do you know that n(t) is a linear function?
c. determine the independent and dependent variable.
d. find n(4). explain what this result means.

Explanation:

Step1: Find the function for part A

The initial number of unwatched shows is 85 and it increases by 20 per week. Using the slope - intercept form $y = mx + b$, where $m$ is the rate of change and $b$ is the initial value, we get $N(t)=85 + 20t$.

Step2: Justify linearity for part B

A linear function has a constant rate of change. Since the number of unwatched shows increases by a fixed amount (20) each week, the function is linear.

Step3: Identify variables for part C

The independent variable is the input of the function (time $t$), and the dependent variable is the output (number of unwatched shows $N$).

Step4: Calculate $N(4)$ for part D

Substitute $t = 4$ into $N(t)=85 + 20t$. First, multiply 20 by 4 to get 80, then add 85 to 80 to get 165. This represents the number of unwatched shows after 4 weeks.

Answer:

A. $N(t)=85 + 20t$
B. The rate of change is constant. The number of unwatched shows increases by 20 per week, which means the slope of the function is a constant value (20). In a linear - function, the general form is $y=mx + b$ where $m$ (the slope) is constant. Here, $m = 20$ and $b = 85$, so $N(t)$ is linear.
C. The independent variable is $t$ (time in weeks) because it is the variable that we can change or control. The dependent variable is $N$ (the number of unwatched shows) because its value depends on the value of $t$.
D. To find $N(4)$, substitute $t = 4$ into the function $N(t)=85+20t$.
\[

$$\begin{align*} N(4)&=85 + 20\times4\\ &=85+80\\ &=165 \end{align*}$$

\]
This result means that after 4 weeks, there are 165 unwatched shows in Sylvia's queue.