To find [tex]\( v(140) \)[/tex], we need to determine the amount of time Ryan is allowed to play video games when he spends 140 minutes on homework and chores. The table provided gives a list of values for [tex]\( w \)[/tex] (the minutes spent on homework and chores) and the corresponding [tex]\( v(w) \)[/tex] (the minutes allowed for video games).
Let's look at the table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
w & 20 & 40 & 60 & 80 & 100 & 120 & 140 \\
\hline
v(w) & 35 & 70 & 105 & 140 & 175 & 210 & 245 \\
\hline
\end{array}
\][/tex]
To find [tex]\( v(140) \)[/tex], we locate the value in the table where [tex]\( w = 140 \)[/tex]. According to the table, when [tex]\( w = 140 \)[/tex]:
[tex]\[
v(140) = 245
\][/tex]
This means Ryan is allowed to play video games for 245 minutes if he spends 140 minutes on homework and chores.
Interpretation:
The relationship between the time spent on homework and chores [tex]\( w \)[/tex] and the time allowed for video games [tex]\( v(w) \)[/tex] shows a consistent pattern. In this case, when Ryan dedicates 140 minutes to homework and chores, his parents allow him to enjoy 245 minutes of video game time. This indicates a direct relationship: more time spent on homework and chores results in more time permitted for video games.
