Answer :
To determine which combinations of new-release and classic DVDs the town librarian could purchase without exceeding a budget of $500, we need to calculate the total cost for each combination and see if it stays within the budget constraint.
The cost of a new-release movie is [tex]$20 each, and the cost of a classic movie is $[/tex]8 each. Let's examine each combination:
1. Combination (x=8, y=45):
- Number of new-release DVDs: \( x = 8 \)
- Number of classic DVDs: \( y = 45 \)
- Cost of new-release DVDs: \( 8 \times 20 = 160 \) dollars
- Cost of classic DVDs: \( 45 \times 8 = 360 \) dollars
- Total cost: \( 160 + 360 = 520 \) dollars
2. Combination (x=10, y=22):
- Number of new-release DVDs: \( x = 10 \)
- Number of classic DVDs: \( y = 22 \)
- Cost of new-release DVDs: \( 10 \times 20 = 200 \) dollars
- Cost of classic DVDs: \( 22 \times 8 = 176 \) dollars
- Total cost: \( 200 + 176 = 376 \) dollars
3. Combination (x=16, y=22):
- Number of new-release DVDs: \( x = 16 \)
- Number of classic DVDs: \( y = 22 \)
- Cost of new-release DVDs: \( 16 \times 20 = 320 \) dollars
- Cost of classic DVDs: \( 22 \times 8 = 176 \) dollars
- Total cost: \( 320 + 176 = 496 \) dollars
4. Combination (x=18, y=18):
- Number of new-release DVDs: \( x = 18 \)
- Number of classic DVDs: \( y = 18 \)
- Cost of new-release DVDs: \( 18 \times 20 = 360 \) dollars
- Cost of classic DVDs: \( 18 \times 8 = 144 \) dollars
- Total cost: \( 360 + 144 = 504 \) dollars
Summary:
- Combination (x=8, y=45) results in a total cost of 520 dollars, which exceeds the budget.
- Combination (x=10, y=22) results in a total cost of 376 dollars, which is within the budget.
- Combination (x=16, y=22) results in a total cost of 496 dollars, which is within the budget.
- Combination (x=18, y=18) results in a total cost of 504 dollars, which exceeds the budget.
Thus, the combinations within the budget are:
- \( (x=10, y=22) \)
- \( (x=16, y=22) \)
These values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] represent the number of new-release and classic movies the librarian could buy without exceeding the budget of $500.
The cost of a new-release movie is [tex]$20 each, and the cost of a classic movie is $[/tex]8 each. Let's examine each combination:
1. Combination (x=8, y=45):
- Number of new-release DVDs: \( x = 8 \)
- Number of classic DVDs: \( y = 45 \)
- Cost of new-release DVDs: \( 8 \times 20 = 160 \) dollars
- Cost of classic DVDs: \( 45 \times 8 = 360 \) dollars
- Total cost: \( 160 + 360 = 520 \) dollars
2. Combination (x=10, y=22):
- Number of new-release DVDs: \( x = 10 \)
- Number of classic DVDs: \( y = 22 \)
- Cost of new-release DVDs: \( 10 \times 20 = 200 \) dollars
- Cost of classic DVDs: \( 22 \times 8 = 176 \) dollars
- Total cost: \( 200 + 176 = 376 \) dollars
3. Combination (x=16, y=22):
- Number of new-release DVDs: \( x = 16 \)
- Number of classic DVDs: \( y = 22 \)
- Cost of new-release DVDs: \( 16 \times 20 = 320 \) dollars
- Cost of classic DVDs: \( 22 \times 8 = 176 \) dollars
- Total cost: \( 320 + 176 = 496 \) dollars
4. Combination (x=18, y=18):
- Number of new-release DVDs: \( x = 18 \)
- Number of classic DVDs: \( y = 18 \)
- Cost of new-release DVDs: \( 18 \times 20 = 360 \) dollars
- Cost of classic DVDs: \( 18 \times 8 = 144 \) dollars
- Total cost: \( 360 + 144 = 504 \) dollars
Summary:
- Combination (x=8, y=45) results in a total cost of 520 dollars, which exceeds the budget.
- Combination (x=10, y=22) results in a total cost of 376 dollars, which is within the budget.
- Combination (x=16, y=22) results in a total cost of 496 dollars, which is within the budget.
- Combination (x=18, y=18) results in a total cost of 504 dollars, which exceeds the budget.
Thus, the combinations within the budget are:
- \( (x=10, y=22) \)
- \( (x=16, y=22) \)
These values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] represent the number of new-release and classic movies the librarian could buy without exceeding the budget of $500.
