The town librarian bought a combination of new-release movies on DVD for [tex]\$20[/tex] and classic movies on DVD for [tex]\$8[/tex]. Let [tex]x[/tex] represent the number of new releases, and let [tex]y[/tex] represent the number of classics. If the librarian had a budget of [tex]\$500[/tex] and wanted to purchase as many DVDs as possible, which values of [tex]x[/tex] and [tex]y[/tex] could represent the number of new-release and classic movies bought?

A. [tex]x=8, y=45[/tex]

B. [tex]x=10, y=22[/tex]

C. [tex]x=16, y=22[/tex]

D. [tex]x=18, y=18[/tex]



Answer :

To determine which values of [tex]\( x \)[/tex] (number of new-release movies) and [tex]\( y \)[/tex] (number of classic movies) the librarian could have bought within a budget of \[tex]$500, we need to calculate the total cost for each combination provided and check if it fits into the budget. The cost formulas are: - New-release movie: \(\$[/tex]20 \times x\)
- Classic movie: [tex]\(\$8 \times y\)[/tex]

1. For the combination [tex]\( (x = 8, y = 45) \)[/tex]:
[tex]\[ \text{Total Cost} = 20 \times 8 + 8 \times 45 \][/tex]
[tex]\[ \text{Total Cost} = 160 + 360 = 520 \][/tex]
Since \[tex]$520 exceeds the \$[/tex]500 budget, this combination is not valid.

2. For the combination [tex]\( (x = 10, y = 22) \)[/tex]:
[tex]\[ \text{Total Cost} = 20 \times 10 + 8 \times 22 \][/tex]
[tex]\[ \text{Total Cost} = 200 + 176 = 376 \][/tex]
Since \[tex]$376 is within the \$[/tex]500 budget, this combination is valid.

3. For the combination [tex]\( (x = 16, y = 22) \)[/tex]:
[tex]\[ \text{Total Cost} = 20 \times 16 + 8 \times 22 \][/tex]
[tex]\[ \text{Total Cost} = 320 + 176 = 496 \][/tex]
Since \[tex]$496 is within the \$[/tex]500 budget, this combination is also valid.

4. For the combination [tex]\( (x = 18, y = 18) \)[/tex]:
[tex]\[ \text{Total Cost} = 20 \times 18 + 8 \times 18 \][/tex]
[tex]\[ \text{Total Cost} = 360 + 144 = 504 \][/tex]
Since \[tex]$504 exceeds the \$[/tex]500 budget, this combination is not valid.

Hence, the valid combinations of the number of new-release and classic movies bought within the \$500 budget are:
[tex]\[ (x = 10, y = 22) \quad \text{and} \quad (x = 16, y = 22) \][/tex]