Question

leonardo has decided to open up a hotdog stand to raise money for charity. the amount of money leonardo makes from their hotdog stand is modeled by the function $f(x)$ where $x$ is the number of hours after the hotdog stand opens. find the average rate of change of the linear function $f(x)$ in the table from $x = 5$ to $x = 7$. \

$$\begin{tabular}{|c|c|} \\hline $x$ & $f(x)$ \\\\ \\hline 1 & 11 \\\\ \\hline 4 & 71 \\\\ \\hline 5 & 103 \\\\ \\hline 7 & 185 \\\\ \\hline 9 & 291 \\\\ \\hline \\end{tabular}$$

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) from \( x = a \) to \( x = b \) is given by \( \frac{f(b)-f(a)}{b - a} \). Here, \( a = 5 \), \( b = 7 \), so we need to find \( f(5) \) and \( f(7) \) from the table.

Step2: Identify \( f(5) \) and \( f(7) \) from the table

From the table, when \( x = 5 \), \( f(5)=103 \), and when \( x = 7 \), \( f(7)=185 \).

Step3: Substitute into the average rate of change formula

Substitute \( a = 5 \), \( b = 7 \), \( f(5)=103 \), and \( f(7)=185 \) into the formula:
\[
\frac{f(7)-f(5)}{7 - 5}=\frac{185 - 103}{7 - 5}
\]

Step4: Simplify the numerator and the denominator

First, calculate the numerator: \( 185-103 = 82 \). Then, calculate the denominator: \( 7 - 5=2 \). Now, divide the numerator by the denominator: \( \frac{82}{2}=41 \).

Answer:

41