To determine which combinations of new-release and classic DVDs the librarian could buy within the given budget of [tex]$500, let's examine the total cost for each combination given the cost of new-release ($[/tex]20) and classic ($8) DVDs.
1. Combination 1: [tex]\( x = 8, y = 45 \)[/tex]
- Total cost [tex]\( = 8 \times 20 + 45 \times 8 \)[/tex]
- Calculate:
[tex]\[
= 160 + 360 = 520
\][/tex]
- Since 520 exceeds the budget of 500, this combination is not feasible.
2. Combination 2: [tex]\( x = 10, y = 22 \)[/tex]
- Total cost [tex]\( = 10 \times 20 + 22 \times 8 \)[/tex]
- Calculate:
[tex]\[
= 200 + 176 = 376
\][/tex]
- Since 376 is within the budget of 500, this combination is feasible.
3. Combination 3: [tex]\( x = 16, y = 22 \)[/tex]
- Total cost [tex]\( = 16 \times 20 + 22 \times 8 \)[/tex]
- Calculate:
[tex]\[
= 320 + 176 = 496
\][/tex]
- Since 496 is within the budget of 500, this combination is feasible.
4. Combination 4: [tex]\( x = 18, y = 18 \)[/tex]
- Total cost: [tex]\( 18 \times 20 + 18 \times 8 \)[/tex]
- Calculate:
[tex]\[
= 360 + 144 = 504
\][/tex]
- Since 504 exceeds the budget of 500, this combination is not feasible.
Thus, the feasible combinations of the number of new-release and classic DVDs within the given budget are:
- [tex]\( x = 10, y = 22 \)[/tex]
- [tex]\( x = 16, y = 22 \)[/tex]
