How many cat pens and dog runs can Carlos and Clarita afford to purchase given their space constraints of 360 ft² and the total start-up costs of $1,280 if cat pens cost $32 each a... How many cat pens and dog runs can Carlos and Clarita afford to purchase given their space constraints of 360 ft² and the total start-up costs of $1,280 if cat pens cost $32 each and dog runs cost $80 each?
Understand the Problem
The question involves calculating how many cat pens and dog runs Carlos and Clarita can afford based on their space and budget constraints. We need to consider both the area occupied by the pens/runs and the costs associated with them.
Answer
The feasible maximum combinations for cat pens and dog runs are \( (20, 0) \) or \( (0, 15) \).
Answer for screen readers
The maximum number of cat pens ( x ) and dog runs ( y ) they can afford while satisfying all constraints is ( (x, y) = (20, 0) ) or ( (x, y) = (0, 15) ), which translates to either 20 cat pens and no dog runs or no cat pens and 15 dog runs.
Steps to Solve
- Define Variables for Cat Pens and Dog Runs
Let ( x ) be the number of cat pens and ( y ) be the number of dog runs.
- Set Up the Area Constraint
The total area occupied by the cat pens and dog runs should be less than or equal to 360 ft².
Each cat pen requires 6 ft² and each dog run requires 24 ft², so we can formulate the inequality:
$$ 6x + 24y \leq 360 $$
- Set Up the Cost Constraint
Carlos and Clarita need to manage their budget of $1,280 for purchasing cat pens and dog runs. The cost for each cat pen is $32, and for each dog run, it's $80. The financial inequality can be expressed as:
$$ 32x + 80y \leq 1280 $$
- Simplify the Inequalities
To simplify the inequalities for easier graphing, we can divide both sides of the area constraint by 6 and the cost constraint by 16:
- Area constraint becomes:
$$ x + 4y \leq 60 $$
- Cost constraint becomes:
$$ 2x + 5y \leq 80 $$
- Graph the Constraints
Now plot the two inequalities on a graph. The feasible region is where both inequalities overlap. Find the intersection points by solving the system of equations formed by setting the left sides equal to the right:
- For ( x + 4y = 60 )
- For ( 2x + 5y = 80 )
- Find the Intersection Points
Solve the system of equations:
- From ( x + 4y = 60 ), express ( x = 60 - 4y )
- Substitute in ( 2(60 - 4y) + 5y = 80 ) to find ( y )
- Determine ( x ) using the value of ( y )
- Test Feasible Solutions
Evaluate the feasible integer combinations of ( x ) and ( y ) that satisfy both constraints and do not exceed the given area and budget.
The maximum number of cat pens ( x ) and dog runs ( y ) they can afford while satisfying all constraints is ( (x, y) = (20, 0) ) or ( (x, y) = (0, 15) ), which translates to either 20 cat pens and no dog runs or no cat pens and 15 dog runs.
More Information
Carlos and Clarita need to balance their space and budget while maximizing the number of animals they can care for. This is a classic example of using linear inequalities to find feasible solutions in real-life constraints.
Tips
- Failing to check if solutions meet both constraints.
- Not considering integer solutions when calculating cat pens and dog runs.
- Forgetting to simplify inequalities before graphing.
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