Asked by Ava
How Do I solve this system of linear equalities word problem?
"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."
"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."
Answers
Answered by
Reiny
number of cups of lemonade --- x
number of cupcakes ---- y
2x + y ≥ 25, where x > 5, y > 5
x .. y
5 15
6 13
7 11
...
number of cupcakes ---- y
2x + y ≥ 25, where x > 5, y > 5
x .. y
5 15
6 13
7 11
...
Answered by
oobleck
Let
x = # cups of lemonade
y = # cupcakes
What have they told you?
x >= 5
y >= 5
2x + 1y >= 25
So, start with the smallest value of x, and then let it grow.
Start with the smallest acceptable profit: 25
x y profit
5 15 25
6 13 25
...
Now, raise the profit to 26 and do it again
5 16 26
6 14 26
...
Or, graph all three of those lines, and shade the solution sets.
Then just pick any points in the shaded area.
x = # cups of lemonade
y = # cupcakes
What have they told you?
x >= 5
y >= 5
2x + 1y >= 25
So, start with the smallest value of x, and then let it grow.
Start with the smallest acceptable profit: 25
x y profit
5 15 25
6 13 25
...
Now, raise the profit to 26 and do it again
5 16 26
6 14 26
...
Or, graph all three of those lines, and shade the solution sets.
Then just pick any points in the shaded area.
Answered by
Bosnian
C = cupcakes
p = profit
Sandy makes $2 profit on every cup of lemonade that she sells and $1 on every cupcake that she sells neans:
p = $2 L + $1C
p = 2 L + C
She wants to earn at least $25 per day means:
p ≥ 25
2 L + C ≥ 25
Sandy wants to sell at least 5 cups of lemonade and at least 5 cupcakes per day means:
L ≥ 5
C ≥ 5
Start with L = 5
2 L + C ≥ 25
2 ∙ 5 + C ≥ 25
10 + C ≥ 25
C ≥ 15
L = 6
2 L + C ≥ 25
2 ∙ 6 + C ≥ 25
12 + C ≥ 25
C ≥ 13
L = 7
2 L + C ≥ 25
2 ∙ 7 + C ≥ 25
14 + C ≥ 25
C ≥ 11
L = 8
2 L + C ≥ 25
2 ∙ 8 + C ≥ 25
16 + C ≥ 25
C ≥ 9
L = 9
2 L + C ≥ 25
2 ∙ 9 + C ≥ 25
18 + C ≥ 25
C ≥ 7
L = 10
2 L + C ≥ 25
2 ∙ 10 + C ≥ 25
20 + C ≥ 25
C ≥ 5
L = 11
2 L + C ≥ 25
2 ∙ 11 + C ≥ 25
22 + C ≥ 25
C ≥ 3
L = 12
2 L + C ≥ 25
2 ∙ 12 + C ≥ 25
24 + C ≥ 25
C ≥ 1
For L = 13
2 L + C ≥ 25
2 ∙ 13 + C ≥ 25
26 + C ≥ 25
C ≥ - 1
For L ≥ 13 , C is negative so you must reject cobinations L ≥ 13 C. In other words, for L ≥ 13 the lemonade sold would earn more than $25, so she wouldn't even sell cupcakes.
So possible combinations are:
L = 5 , C ≥ 15
L = 6 , C ≥ 13
L = 7 , C ≥ 11
L = 8 , C ≥ 9
L = 9 , C ≥ 7
L = 10 , C ≥ 5
L = 11 , C ≥ 3
L = 12 , C ≥ 1
This is the MINIMUM number of combinations that earns the required profit.
Of course she can sell more than this.
p = profit
Sandy makes $2 profit on every cup of lemonade that she sells and $1 on every cupcake that she sells neans:
p = $2 L + $1C
p = 2 L + C
She wants to earn at least $25 per day means:
p ≥ 25
2 L + C ≥ 25
Sandy wants to sell at least 5 cups of lemonade and at least 5 cupcakes per day means:
L ≥ 5
C ≥ 5
Start with L = 5
2 L + C ≥ 25
2 ∙ 5 + C ≥ 25
10 + C ≥ 25
C ≥ 15
L = 6
2 L + C ≥ 25
2 ∙ 6 + C ≥ 25
12 + C ≥ 25
C ≥ 13
L = 7
2 L + C ≥ 25
2 ∙ 7 + C ≥ 25
14 + C ≥ 25
C ≥ 11
L = 8
2 L + C ≥ 25
2 ∙ 8 + C ≥ 25
16 + C ≥ 25
C ≥ 9
L = 9
2 L + C ≥ 25
2 ∙ 9 + C ≥ 25
18 + C ≥ 25
C ≥ 7
L = 10
2 L + C ≥ 25
2 ∙ 10 + C ≥ 25
20 + C ≥ 25
C ≥ 5
L = 11
2 L + C ≥ 25
2 ∙ 11 + C ≥ 25
22 + C ≥ 25
C ≥ 3
L = 12
2 L + C ≥ 25
2 ∙ 12 + C ≥ 25
24 + C ≥ 25
C ≥ 1
For L = 13
2 L + C ≥ 25
2 ∙ 13 + C ≥ 25
26 + C ≥ 25
C ≥ - 1
For L ≥ 13 , C is negative so you must reject cobinations L ≥ 13 C. In other words, for L ≥ 13 the lemonade sold would earn more than $25, so she wouldn't even sell cupcakes.
So possible combinations are:
L = 5 , C ≥ 15
L = 6 , C ≥ 13
L = 7 , C ≥ 11
L = 8 , C ≥ 9
L = 9 , C ≥ 7
L = 10 , C ≥ 5
L = 11 , C ≥ 3
L = 12 , C ≥ 1
This is the MINIMUM number of combinations that earns the required profit.
Of course she can sell more than this.
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