Answer :
Let's break down each aspect to determine whether they vary directly or not.
1. The gallons of gasoline used and the number of miles driven:
- Statement: "Driving twice as many miles will use twice as much gas."
- Analysis: This implies that the gallons of gasoline vary directly with the number of miles driven. Mathematically, if you drive twice as many miles, you use twice as much gas, indicating a direct proportionality.
- Conclusion: Yes, they vary directly.
2. The volume of a circular hotel swimming pool and the square of its radius:
- Statement: "In the equation for the volume of a cylinder [tex]\( V = \pi hr^2 \)[/tex], doubling the value of [tex]\( r^2 \)[/tex] will double the value of [tex]\( V \)[/tex]."
- Analysis: The volume [tex]\( V \)[/tex] of the swimming pool is given by [tex]\( V = \pi hr^2 \)[/tex], where [tex]\( h \)[/tex] and [tex]\( \pi \)[/tex] are constants. Since [tex]\( V \)[/tex] is directly proportional to [tex]\( r^2 \)[/tex], doubling [tex]\( r^2 \)[/tex] results in doubling [tex]\( V \)[/tex].
- Conclusion: Yes, they vary directly.
3. The number of attractions visited in a day and the time spent at each one:
- Statement: "Doubling the time spent at attractions will not result in being able to visit twice as many."
- Analysis: Here, increasing the time spent at each attraction does not directly influence the number of attractions visited. Instead, if you spend more time at each attraction, you might visit fewer attractions.
- Conclusion: No, they do not vary directly.
4. The number of pins that can be bought for [tex]$20 and the price per pin: - Statement: "When the price of the pins is doubled, the number that can be purchased is halved." - Analysis: Here, the relationship between the number of pins and their price is inversely proportional. Doubling the price per pin results in buying half the number of pins. - Conclusion: No, they do not vary directly. Therefore, the final table should be: \[ \begin{tabular}{|c|c|} \hline \text{Aspect} & \text{Vary Directly?} \\ \hline \begin{tabular}{l} the gallons of gasoline used and the number of miles driven \\ (Driving twice as many miles will use twice as much gas.) \end{tabular} & Yes \\ \hline \begin{tabular}{l} the volume of a circular hotel swimming pool and the square of its radius \\ (In the equation for the volume of a cylinder, \( V = \pi hr^2 \), doubling the value \\ of \( r^2 \) will \\ $[/tex]\qquad[tex]$ \\ the value of \( V \).) \end{tabular} & Yes \\ \hline \begin{tabular}{l} the number of attractions visited in a day and the time spent at each one \\ (Doubling the time spent at attractions will not result in being able to visit \\ twice as many.) \end{tabular} & No \\ \hline \begin{tabular}{l} the number of pins that can be bought for \( \$[/tex] 20 \) and the price per pin \\
(When the price of the pins is doubled, the number that can be purchased \\
halves.)
\end{tabular} & No \\
\hline
\end{tabular}
\]
1. The gallons of gasoline used and the number of miles driven:
- Statement: "Driving twice as many miles will use twice as much gas."
- Analysis: This implies that the gallons of gasoline vary directly with the number of miles driven. Mathematically, if you drive twice as many miles, you use twice as much gas, indicating a direct proportionality.
- Conclusion: Yes, they vary directly.
2. The volume of a circular hotel swimming pool and the square of its radius:
- Statement: "In the equation for the volume of a cylinder [tex]\( V = \pi hr^2 \)[/tex], doubling the value of [tex]\( r^2 \)[/tex] will double the value of [tex]\( V \)[/tex]."
- Analysis: The volume [tex]\( V \)[/tex] of the swimming pool is given by [tex]\( V = \pi hr^2 \)[/tex], where [tex]\( h \)[/tex] and [tex]\( \pi \)[/tex] are constants. Since [tex]\( V \)[/tex] is directly proportional to [tex]\( r^2 \)[/tex], doubling [tex]\( r^2 \)[/tex] results in doubling [tex]\( V \)[/tex].
- Conclusion: Yes, they vary directly.
3. The number of attractions visited in a day and the time spent at each one:
- Statement: "Doubling the time spent at attractions will not result in being able to visit twice as many."
- Analysis: Here, increasing the time spent at each attraction does not directly influence the number of attractions visited. Instead, if you spend more time at each attraction, you might visit fewer attractions.
- Conclusion: No, they do not vary directly.
4. The number of pins that can be bought for [tex]$20 and the price per pin: - Statement: "When the price of the pins is doubled, the number that can be purchased is halved." - Analysis: Here, the relationship between the number of pins and their price is inversely proportional. Doubling the price per pin results in buying half the number of pins. - Conclusion: No, they do not vary directly. Therefore, the final table should be: \[ \begin{tabular}{|c|c|} \hline \text{Aspect} & \text{Vary Directly?} \\ \hline \begin{tabular}{l} the gallons of gasoline used and the number of miles driven \\ (Driving twice as many miles will use twice as much gas.) \end{tabular} & Yes \\ \hline \begin{tabular}{l} the volume of a circular hotel swimming pool and the square of its radius \\ (In the equation for the volume of a cylinder, \( V = \pi hr^2 \), doubling the value \\ of \( r^2 \) will \\ $[/tex]\qquad[tex]$ \\ the value of \( V \).) \end{tabular} & Yes \\ \hline \begin{tabular}{l} the number of attractions visited in a day and the time spent at each one \\ (Doubling the time spent at attractions will not result in being able to visit \\ twice as many.) \end{tabular} & No \\ \hline \begin{tabular}{l} the number of pins that can be bought for \( \$[/tex] 20 \) and the price per pin \\
(When the price of the pins is doubled, the number that can be purchased \\
halves.)
\end{tabular} & No \\
\hline
\end{tabular}
\]
