Question

exercise #4: skylar is selling glasses of lemonade. the function g(t)=t+4 / 2 models the number of glasses she has sold, g, after t - hours. what is the average rate at which she is selling lemonade between t = 2 and t = 4 hours? include proper units in your answer.

Explanation:

Step1: Recall average - rate formula

The average rate of change of a function $y = g(t)$ over the interval $[a,b]$ is given by $\frac{g(b)-g(a)}{b - a}$. Here, $a = 2$, $b=4$, and $g(t)=\frac{t}{t + 4}$.

Step2: Calculate $g(4)$ and $g(2)$

First, find $g(4)$:
$g(4)=\frac{4}{4 + 4}=\frac{4}{8}=\frac{1}{2}$
Then, find $g(2)$:
$g(2)=\frac{2}{2+4}=\frac{2}{6}=\frac{1}{3}$

Step3: Calculate the average - rate of change

The average rate of change over the interval $[2,4]$ is $\frac{g(4)-g(2)}{4 - 2}$.
Substitute the values of $g(4)$ and $g(2)$:
\[

$$\begin{align*} \frac{g(4)-g(2)}{4 - 2}&=\frac{\frac{1}{2}-\frac{1}{3}}{2}\\ &=\frac{\frac{3 - 2}{6}}{2}\\ &=\frac{\frac{1}{6}}{2}\\ &=\frac{1}{12} \end{align*}$$

\]

Answer:

$\frac{1}{12}$