Question

sandy makes $2 profit on every cup of lemonade that she sells and $1 on every cupcake that she sells. sandy wants to sell at least 5 cups of lemonade and at least 5 cupcakes per day. she wants to earn at least $25 per day. show all the possible combinations of lemonade and cupcakes that sandy needs to sell in order to meet her goals. then list two possible combinations.

Explanation:

Step1: Define Variables

Let \( x \) be the number of lemonade cups, \( y \) be the number of cupcakes.

Step2: Write Constraints

• Profit: \( 2x + y \geq 25 \)
• Lemonade: \( x \geq 5 \)
• Cupcakes: \( y \geq 5 \)

Step3: Find Boundaries

For \( x = 5 \), \( 2(5)+y \geq 25 \Rightarrow y \geq 15 \).
For \( y = 5 \), \( 2x + 5 \geq 25 \Rightarrow x \geq 10 \).

Step4: Identify Region

The feasible region is where \( x \geq 5 \), \( y \geq 5 \), and \( 2x + y \geq 25 \).

Step5: List Combinations

• Combo 1: \( x = 10 \), \( y = 5 \) (Check: \( 2(10)+5 = 25 \geq 25 \), \( 10 \geq 5 \), \( 5 \geq 5 \))
• Combo 2: \( x = 5 \), \( y = 15 \) (Check: \( 2(5)+15 = 25 \geq 25 \), \( 5 \geq 5 \), \( 15 \geq 5 \))

Answer:

Feasible region: \( x \geq 5 \), \( y \geq 5 \), \( 2x + y \geq 25 \).
Two combinations: (10, 5) [10 lemonade, 5 cupcakes], (5, 15) [5 lemonade, 15 cupcakes].