To determine which combination of new-release movies (x) and classic movies (y) fits the librarian's budget of [tex]$500, we need to calculate the total cost for each given pair \((x, y)\), and then check if the total cost is within the budget of $[/tex]500.
The cost formula can be represented as:
[tex]\[ \text{Total Cost} = 20x + 8y \][/tex]
Let's calculate the costs for each combination:
1. For [tex]\( x = 8 \)[/tex] and [tex]\( y = 45 \)[/tex]:
[tex]\[
\text{Total Cost} = 20 \cdot 8 + 8 \cdot 45 = 160 + 360 = 520
\][/tex]
The total cost is [tex]$520, which is above the budget of $[/tex]500. Hence, this combination is not within the budget.
2. For [tex]\( x = 10 \)[/tex] and [tex]\( y = 22 \)[/tex]:
[tex]\[
\text{Total Cost} = 20 \cdot 10 + 8 \cdot 22 = 200 + 176 = 376
\][/tex]
The total cost is [tex]$376, which is within the budget of $[/tex]500. Hence, this combination is within the budget.
3. For [tex]\( x = 16 \)[/tex] and [tex]\( y = 22 \)[/tex]:
[tex]\[
\text{Total Cost} = 20 \cdot 16 + 8 \cdot 22 = 320 + 176 = 496
\][/tex]
The total cost is [tex]$496, which is within the budget of $[/tex]500. Hence, this combination is within the budget.
4. For [tex]\( x = 18 \)[/tex] and [tex]\( y = 18 \)[/tex]:
[tex]\[
\text{Total Cost} = 20 \cdot 18 + 8 \cdot 18 = 360 + 144 = 504
\][/tex]
The total cost is [tex]$504, which is above the budget of $[/tex]500. Hence, this combination is not within the budget.
Summarizing the results:
- The combination [tex]\( (x = 8, y = 45) \)[/tex] is above the budget.
- The combination [tex]\( (x = 10, y = 22) \)[/tex] is within the budget.
- The combination [tex]\( (x = 16, y = 22) \)[/tex] is within the budget.
- The combination [tex]\( (x = 18, y = 18) \)[/tex] is above the budget.
Therefore, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that meet the budget constraint are:
[tex]\[ x = 10, y = 22 \][/tex]
and
[tex]\[ x = 16, y = 22 \][/tex]
