To determine which statement best describes the function that represents the situation where the final cost of a sale item is determined by multiplying the price on the tag by [tex]\(75\%\)[/tex], let's carefully analyze the provided tabular data and the relationship between the tag price and the final cost:
[tex]\[
\begin{tabular}{|c|c|}
\hline
Price on the Tag, $x$ & Final Cost $0.75x$ \\
\hline
\$ 10 & $0.75(10) = \$7.50$ \\
\hline
\$ 20 & $0.75(20) = \$15.00$ \\
\hline
\$ 30 & $0.75(30) = \$22.50$ \\
\hline
\$ 40 & $0.75(40) = \$30.00$ \\
\hline
\end{tabular}
\][/tex]
Next, we will determine if the relationship is linear by checking if the ratio of the final cost to the tag price is consistent across all given prices:
1. For a price tag of \[tex]$10, the final cost is \(0.75 \cdot 10 = \$[/tex]7.50\). The ratio is [tex]\(\frac{7.50}{10} = 0.75\)[/tex].
2. For a price tag of \[tex]$20, the final cost is \(0.75 \cdot 20 = \$[/tex]15.00\). The ratio is [tex]\(\frac{15.00}{20} = 0.75\)[/tex].
3. For a price tag of \[tex]$30, the final cost is \(0.75 \cdot 30 = \$[/tex]22.50\). The ratio is [tex]\(\frac{22.50}{30} = 0.75\)[/tex].
4. For a price tag of \[tex]$40, the final cost is \(0.75 \cdot 40 = \$[/tex]30.00\). The ratio is [tex]\(\frac{30.00}{40} = 0.75\)[/tex].
Since the ratio of the final cost to the tag price is constant at [tex]\(0.75\)[/tex] for all given prices, it indicates that the relationship between the tag price and the final cost is linear.
Thus, the correct statement is:
"It is linear because the ratio of the change in the final cost compared to the rate of change in the price tag is constant."
