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 and describe all the possible combinations of lemonade and cupcakes that Sandy needs to sell to meet her goals. List two possible combinations.

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Q. 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 and describe all the possible combinations of lemonade and cupcakes that Sandy needs to sell to meet her goals. List two possible combinations.

Define Variables and Constraints: Define the variables and the constraints. $\newline$ Let $L$ be the number of cups of lemonade Sandy sells. $\newline$ Let $C$ be the number of cupcakes Sandy sells. $\newline$ Sandy makes $\$2$ profit on every cup of lemonade and $\$1$ on every cupcake. $\newline$ Sandy's goal is to earn at least $\$25$ per day. $\newline$ Sandy wants to sell at least $5$ cups of lemonade and at least $5$ cupcakes per day. $\newline$ The constraints can be written as: $\newline$ $L \geq 5$ $\newline$ $C \geq 5$ $\newline$ $2L + C \geq 25$
Minimum Lemonade Cups: Find the minimum number of lemonade cups Sandy needs to sell to meet the goal without selling any cupcakes. $\newline$ Since Sandy makes $\$2$ profit on every cup of lemonade, we can calculate the minimum number of lemonade cups needed to reach $\$25$ by dividing the goal by the profit per cup of lemonade. $\newline$ Minimum lemonade cups = $\frac{\$25}{\$2}$ per cup = $12.5$ cups $\newline$ Since Sandy cannot sell half a cup, she needs to sell at least $13$ cups of lemonade to meet the goal without selling any cupcakes.
Minimum Cupcakes: Find the minimum number of cupcakes Sandy needs to sell to meet the goal without selling any lemonade. $\newline$ Since Sandy makes $\$1$ profit on every cupcake, we can calculate the minimum number of cupcakes needed to reach $\$25$ by dividing the goal by the profit per cupcake. $\newline$ Minimum cupcakes = $\frac{\$25}{\$1}$ per cupcake = $25$ cupcakes $\newline$ Sandy needs to sell at least $25$ cupcakes to meet the goal without selling any lemonade.
Possible Combinations: List two possible combinations that meet the constraints. $\newline$ Combination $1$ : Sandy sells $5$ cups of lemonade and the rest in cupcakes. $\newline$ Using the equation $2L + C \geq 25$ , we substitute $L = 5$ . $\newline$ $2(5) + C \geq 25$ $\newline$ $10 + C \geq 25$ $\newline$ $C \geq 25 - 10$ $\newline$ $C \geq 15$ $\newline$ So, Sandy can sell $5$ cups of lemonade and $15$ cupcakes to meet her goal. $\newline$ Combination $2$ : Sandy sells $6$ cups of lemonade and the rest in cupcakes. $\newline$ Using the equation $2L + C \geq 25$ , we substitute $2L + C \geq 25$ $1$ . $\newline$ $2L + C \geq 25$ $2$ $\newline$ $2L + C \geq 25$ $3$ $\newline$ $2L + C \geq 25$ $4$ $\newline$ $2L + C \geq 25$ $5$ $\newline$ So, Sandy can sell $6$ cups of lemonade and $2L + C \geq 25$ $7$ cupcakes to meet her goal.