Question

Suppose you decide to start a cupcake business. You are able to bake and sell the cupcakes at your own house, so you don't have to pay for a place to run the business. You have $1000 to start the business. Now it is time to purchase the ingredients to get started. In the beginning, you decide to make only one kind of cupcakes. The following recipe shows the ingredients for 12 cupcakes.

Chunky Chocolate Cupcake Recipe
1.5 cup flour
2 teaspoons baking powder
3/4 cup coca powder
1/4 cup vegetable oil
1 cup sugar
1 egg
1/2 cup milk
1 cup butter
4 cups powdered sugar for frosting
1 teaspoon vanilla
Let x be the dozens of cupcakes. The following table shows the cost of the ingredients.

Flour $0.60 per cup
Baking powder $0.20 per teaspoon
Coca powder $1.60 per cup
Vegetable oil $0.80 per cup
Sugar $0.30 per cup
Eggs $0.30 per egg
Milk $0.40 per cup
Butter $0.80 per cup
Powdered sugar $0.60 per cup
Vanilla $1.00 per teaspoon
Let y be the cost, and x be the dozens of cupcakes. Set up a cost function.
Remember, you can spend at most $1000 on the ingredients. Set up an inequality.
Solve your inequality to find out how many dozens of chunky chocolate cupcakes can you make

Answer 1

Amount given to start the business = $1000

Recipe is for baking = 12 cupcakes

Ingredients needed are :

Flour = 1.5 cups; total value = 0.60*1.5 = $0.90

Baking powder = 2 tsp; total value = 0.20*2 = $0.40

Cocoa powder = 3/4 cups; total value = 1.60*3/4 = $1.20

Vegetable oil = 1/4 cup; total value = 0.80*1/4 = $0.20

Sugar = 1 cup; total value = 0.30*1 = $0.30

Egg = 1; total value = 0.30*1 = $0.30

Milk = 1/2 cups; total value = 0.40*1/2 = $0.20

Butter = 1 cup; total value = 0.80*1 = $0.80

Powdered sugar = 4 cups; total value = 0.60*4 = $2.40

Vanilla = 1 tsp; total value = 1*1 = $1.00

Hence, total cost for preparing 12 cupcakes =

0.90+0.40+1.20+0.20+0.30+0.30+0.20+0.80+2.40+1.00 = $7.70

Let y be the cost, and x be the dozens of cupcakes.

So, cost function becomes :

7.70(x)=y

As the cost can at most be $1000, inequality becomes

so
`y\leq 1000`

Solving this we get

`7.70x\leq 1000`

`x\leq (1000)/(7.7)`

`x\leq 129.87`

As cupcakes can be in whole number, so it becomes almost 129 dozen cupcakes.