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 evelyn’s 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 $(0,30)$ and $(15,0)$. Slope $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 30}{15-0}=\frac{- 30}{15}=-2$.

Step2: Write the equation

Using the slope - intercept form $y=mx + b$, where $m=-2$ and $b = 30$ (the $y$-intercept), we get $C=-2t + 30$ or $C = 30 - 2t$.

Step3: Identify the $y$-intercept

The $y$-intercept occurs when $t = 0$. From the graph and the equation, when $t = 0$, $C = 30$, so the $y$-intercept is the point $(0,30)$.

Step4: Interpret the $y$-intercept

In the context of the problem, when $t = 0$ (the day of Evelyn's birthday), the value of $C$ is 30, which means there were 30 chocolates in the box initially.

Answer:

Equation: $C = 30 - 2t$; $y$-intercept: $(0,30)$; Interpretation: The number of chocolates in the box at the start (on Evelyn's birthday, when $t = 0$) is 30.