Let's analyze each option to determine if the number of new release movies, [tex]\( x \)[/tex], and the number of classic movies, [tex]\( y \)[/tex], add up to a total cost that is within the librarian's budget of \[tex]$500.
Option 1: \( x = 8 \), \( y = 45 \)
- The cost for new releases: \( 8 \times 20 = 160 \)
- The cost for classics: \( 45 \times 8 = 360 \)
- Total cost: \( 160 + 360 = 520 \)
Result: Total cost is \( \$[/tex]520 \). This is not within the budget of \[tex]$500.
Option 2: \( x = 10 \), \( y = 22 \)
- The cost for new releases: \( 10 \times 20 = 200 \)
- The cost for classics: \( 22 \times 8 = 176 \)
- Total cost: \( 200 + 176 = 376 \)
Result: Total cost is \( \$[/tex]376 \). This is within the budget of \[tex]$500.
Option 3: \( x = 16 \), \( y = 22 \)
- The cost for new releases: \( 16 \times 20 = 320 \)
- The cost for classics: \( 22 \times 8 = 176 \)
- Total cost: \( 320 + 176 = 496 \)
Result: Total cost is \( \$[/tex]496 \). This is within the budget of \[tex]$500.
Option 4: \( x = 18 \), \( y = 18 \)
- The cost for new releases: \( 18 \times 20 = 360 \)
- The cost for classics: \( 18 \times 8 = 144 \)
- Total cost: \( 360 + 144 = 504 \)
Result: Total cost is \( \$[/tex]504 \). This is not within the budget of \[tex]$500.
Summarizing, the valid options where the total cost is within the budget of \$[/tex]500 are:
- [tex]\( x = 10 \)[/tex], [tex]\( y = 22 \)[/tex]
- [tex]\( x = 16 \)[/tex], [tex]\( y = 22 \)[/tex]
