Math Problem Statement
Bakers' Club is trying to raise funds by selling premium chocolate chip cookies in a school fair. The variable cost to make each cookie is $15.00 and it is being sold for P25.00. So far, the organization has already shelled out #790.00 for the cookie sale. Find the profit function P(x) where x represents the number of cookies sold. Hint: Profit Total Revenue - Total Cost Total Revenue = Price per unit x quantity sold Total Cost = Total variable cost + fixed cost b. If 146 cookies were sold, how much is the total profit? c. How many cookies must be made and sold to break even? Hint: Break even point is the zero of P(x) d. How many cookie should be sold to gain a profit of P250.00?
Solution
Let's break down the problem step by step.
Part a: Finding the Profit Function
To find the profit function , we use the following formulas:
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Total Revenue (TR): This is the total amount earned from selling the cookies. Given that each cookie is sold for , the total revenue is:
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Total Cost (TC): This is the sum of the fixed costs and the total variable cost. The fixed cost is given as . The total variable cost is the cost to make each cookie multiplied by the number of cookies sold, so: Therefore, the total cost is:
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Profit Function (P(x)): Profit is the difference between total revenue and total cost. Substituting the expressions we have: Simplifying this:
Part b: Total Profit for 146 Cookies Sold
To find the profit when 146 cookies are sold, substitute into the profit function :
So, the total profit when 146 cookies are sold is .
Part c: Break-even Point
The break-even point is when the profit is zero, . Set the profit function equal to zero and solve for :
So, 79 cookies must be sold to break even.
Part d: Number of Cookies to Gain a Profit of P250.00
To find the number of cookies needed to achieve a profit of , set and solve for :
So, 104 cookies need to be sold to gain a profit of .
Would you like further details or have any questions?
Here are some related questions:
- What is the meaning of the break-even point in business?
- How would the profit function change if the selling price increased?
- What is the impact of a higher fixed cost on the break-even point?
- How could the Bakers' Club maximize its profit?
- What if the variable cost increases to P20.00 per cookie, how would that affect the profit function?
Tip: Always double-check your fixed and variable costs when calculating the break-even point.
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Math Problem Analysis
Mathematical Concepts
Linear Functions
Profit Analysis
Break-Even Analysis
Formulas
Total Revenue = Price per unit × quantity sold
Total Cost = Fixed Cost + Variable Cost per unit × quantity sold
Profit Function P(x) = Total Revenue - Total Cost
Theorems
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Suitable Grade Level
Grades 10-12