Adam can spend a maximum of $252 on office supplies. Each ream of paper costs $6 and each ink cartridge costs $18. Which of the following graphs represents the possible combination... Adam can spend a maximum of $252 on office supplies. Each ream of paper costs $6 and each ink cartridge costs $18. Which of the following graphs represents the possible combinations of paper and ink cartridges that he may buy?
Understand the Problem
The question is asking which graph represents the possible combinations of paper and ink cartridges that Adam can buy given his budget constraints. It involves analyzing the budget to determine the feasible combinations represented in the graphs.
Answer
Graph C represents the combinations Adam can buy given his budget constraint of $252.
Answer for screen readers
The correct answer is graph C.
Steps to Solve
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Identify Cost Constraints
Given that each ream of paper costs $6 and each ink cartridge costs $18, we can define the total cost equation. Let ( x ) be the number of reams of paper and ( y ) be the number of ink cartridges.
The equation is: $$ 6x + 18y \leq 252 $$
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Simplify the Equation
To make the analysis easier, divide the entire equation by 6: $$ x + 3y \leq 42 $$
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Rearrange the Equation
Rearranging for ( y ) gives us: $$ 3y \leq 42 - x $$ $$ y \leq \frac{42 - x}{3} $$
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Determine Intercepts
Find the intercepts for the graph:
- For ( x )-intercept (set ( y = 0 )): $$ x = 42 $$
- For ( y )-intercept (set ( x = 0 )): $$ 3y = 42 \Rightarrow y = 14 $$
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Graph the Inequality
The line ( y = \frac{42 - x}{3} ) will have an ( x )-intercept at (42, 0) and a ( y )-intercept at (0, 14). The feasible region below this line represents all combinations of paper and ink cartridges within the budget.
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Choose the Correct Graph
Look for the graph that represents the area below the line defined by these intercepts, showing valid combinations of paper and ink cartridges.
The correct answer is graph C.
More Information
Graph C accurately represents the combinations of reams of paper and ink cartridges Adam can buy within his budget. Each point in the shaded region corresponds to a valid purchase combination, satisfying the constraint.
Tips
- Assuming any point above the line is valid: Only points at or below the line satisfy the budget constraint.
- Forgetting to equate the cost correctly or miscalculating the intercepts.
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