To determine which values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] fit within the librarian's budget and maximize the number of DVDs purchased, we can evaluate each combination provided. Remember, [tex]\( x \)[/tex] represents the number of new-release movies at [tex]\( \$20 \)[/tex] each and [tex]\( y \)[/tex] represents the number of classic movies at [tex]\( \$8 \)[/tex] each. The budget is [tex]\( \$500 \)[/tex].
We need to calculate the total expenditure for each combination and check if it is within the budget.
### Combination 1: [tex]\( x = 8 \)[/tex], [tex]\( y = 45 \)[/tex]
[tex]\[
\text{Total Cost} = (8 \times 20) + (45 \times 8)
\][/tex]
[tex]\[
\text{Total Cost} = 160 + 360 = 520
\][/tex]
This combination exceeds the budget of [tex]\( \$500 \)[/tex].
### Combination 2: [tex]\( x = 10 \)[/tex], [tex]\( y = 22 \)[/tex]
[tex]\[
\text{Total Cost} = (10 \times 20) + (22 \times 8)
\][/tex]
[tex]\[
\text{Total Cost} = 200 + 176 = 376
\][/tex]
This combination is within the budget of [tex]\( \$500 \)[/tex].
### Combination 3: [tex]\( x = 16 \)[/tex], [tex]\( y = 22 \)[/tex]
[tex]\[
\text{Total Cost} = (16 \times 20) + (22 \times 8)
\][/tex]
[tex]\[
\text{Total Cost} = 320 + 176 = 496
\][/tex]
This combination is also within the budget of [tex]\( \$500 \)[/tex].
### Combination 4: [tex]\( x = 18 \)[/tex], [tex]\( y = 18 \)[/tex]
[tex]\[
\text{Total Cost} = (18 \times 20) + (18 \times 8)
\][/tex]
[tex]\[
\text{Total Cost} = 360 + 144 = 504
\][/tex]
This combination exceeds the budget of [tex]\( \$500 \)[/tex].
### Conclusion
The combinations that fit within the [tex]\( \$500 \)[/tex] budget are:
- [tex]\( x = 10 \)[/tex], [tex]\( y = 22 \)[/tex] with a total cost of [tex]\( \$376 \)[/tex]
- [tex]\( x = 16 \)[/tex], [tex]\( y = 22 \)[/tex] with a total cost of [tex]\( \$496 \)[/tex]
Thus, the possible combinations that the librarian can afford are:
[tex]\[
(10, 22) \quad \text{and} \quad (16, 22)
\][/tex]
