What are the maximum number of cat pens and dog runs Carlos and Clarita can purchase given the space and cost constraints?

Steps to Solve

1. Define Variables for Cat Pens and Dog Runs

Let ( x ) be the number of cat pens and ( y ) be the number of dog runs.

2. Space Requirement Equation

Each cat pen requires 6 ft² and each dog run requires 24 ft². The total space used must be less than or equal to 360 ft²:

$$ 6x + 24y \leq 360 $$

3. Cost Requirement Equation

The cost of each cat pen is $32 and each dog run is $80. The total cost should be less than or equal to $1,280:

$$ 32x + 80y \leq 1280 $$

4. Setting Up Non-Negativity Constraints

We also need to ensure non-negativity, meaning ( x \geq 0 ) and ( y \geq 0 ).

5. Solving the System of Inequalities

To solve the system:

• Rearranging the space equation gives us:

$$ y \leq \frac{360 - 6x}{24} $$

• Rearranging the cost equation gives:

$$ y \leq \frac{1280 - 32x}{80} $$

6. Finding Intersection Points

Now find intersection points of the lines ( 6x + 24y = 360 ) and ( 32x + 80y = 1280 ) to determine feasible values of ( x ) and ( y ).

7. Graphing Inequalities

You can graph the inequalities and find the feasible region. The vertices of this region will give you the possible values of ( x ) and ( y ).