Question

evelyn was given a box of assorted chocolates for her birthday. each night, evelyn treats herself to some chocolates. let c represent the number of chocolates remaining in the box t days after evelyns birthday. a graph of c is shown below. write an equation for c then state the y - intercept of the graph and determine its interpretation in the context of the problem. note: the line drawn is dotted because the domain of the function is limited to the integers.

Explanation:

Step1: Find the slope

The line passes through points $(0, 30)$ and $(15,0)$. The slope $m$ of a line passing through $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, $x_1 = 0,y_1=30,x_2 = 15,y_2 = 0$. So $m=\frac{0 - 30}{15-0}=\frac{- 30}{15}=-2$.

Step2: Write the equation

The equation of a line in slope - intercept form is $y=mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. Since $m=-2$ and $b = 30$ (from the point $(0,30)$), the equation for $C$ in terms of $t$ is $C=-2t + 30$.

Step3: Identify the y - intercept

The $y$-intercept is the value of $C$ when $t = 0$. From the graph and the equation $C=-2t + 30$, when $t = 0$, $C = 30$.

Step4: Interpret the y - intercept

When $t = 0$ (the day of Evelyn's birthday), the number of chocolates in the box is 30. So the $y$-intercept represents the initial number of chocolates in the box.

Answer:

Equation: $C=-2t + 30$
$y$-intercept: 30
Interpretation: The initial number of chocolates in the box on Evelyn's birthday is 30.