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QuestionMax's soda stand costs: bottles at \$0.32, ingredients at \$0.10, and a \$54 sealer. How many bottles must he sell at \$1.50?

Studdy Solution

STEP 1

Assumptions1. The cost of an empty bottle is 0.32.Thecostofingredientsperbottleis0.32. The cost of ingredients per bottle is 0.103. The cost of the bottle-cap sealer is 544.Thesellingpriceperbottleis544. The selling price per bottle is 1.505. We need to find the number of bottles Max needs to sell to break even (sales equals expenses)

STEP 2

First, we need to find the total cost per bottle. This includes the cost of the empty bottle and the cost of the ingredients.
Cost per bottle=Cost of empty bottle+Cost of ingredientsCost\, per\, bottle = Cost\, of\, empty\, bottle + Cost\, of\, ingredients

STEP 3

Now, plug in the given values for the cost of the empty bottle and the cost of the ingredients to calculate the cost per bottle.
Cost per bottle=$0.32+$0.10Cost\, per\, bottle = \$0.32 + \$0.10

STEP 4

Calculate the cost per bottle.
Cost per bottle=$0.32+$0.10=$0.42Cost\, per\, bottle = \$0.32 + \$0.10 = \$0.42

STEP 5

Now that we have the cost per bottle, we can find the total cost for Max. This includes the cost per bottle and the cost of the bottle-cap sealer.
Total cost=Cost per bottletimesNumber of bottles+Cost of bottle−cap sealerTotal\, cost = Cost\, per\, bottle \\times Number\, of\, bottles + Cost\, of\, bottle-cap\, sealer

STEP 6

We know that Max's sales must equal his expenses to break even. So, we can set up an equation where the total cost equals the total sales.
Total sales=Selling price per bottletimesNumber of bottlesTotal\, sales = Selling\, price\, per\, bottle \\times Number\, of\, bottlesTotal cost=Total salesTotal\, cost = Total\, sales

STEP 7

Substitute the expressions for total cost and total sales from steps5 and6 into the equation from step6.
Cost per bottletimesNumber of bottles+Cost of bottle−cap sealer=Selling price per bottletimesNumber of bottlesCost\, per\, bottle \\times Number\, of\, bottles + Cost\, of\, bottle-cap\, sealer = Selling\, price\, per\, bottle \\times Number\, of\, bottles

STEP 8

Rearrange the equation to solve for the number of bottles.
Number of bottles=Cost of bottle−cap sealerelling price per bottle−Cost per bottleNumber\, of\, bottles = \frac{Cost\, of\, bottle-cap\, sealer}{elling\, price\, per\, bottle - Cost\, per\, bottle}

STEP 9

Plug in the values for the cost of the bottle-cap sealer, the selling price per bottle, and the cost per bottle to calculate the number of bottles.
Number of bottles=$54$.50−$.42Number\, of\, bottles = \frac{\$54}{\$.50 - \$.42}

STEP 10

Calculate the number of bottles Max needs to sell to break even.
Number of bottles=$54$.50−$0.42=60Number\, of\, bottles = \frac{\$54}{\$.50 - \$0.42} =60Max needs to sell60 bottles of soda to break even.

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