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 P(x)P(x)

To find the profit function P(x)P(x), we use the following formulas:

  • Total Revenue (TR): This is the total amount earned from selling the cookies. Total Revenue=Price per unit×quantity sold\text{Total Revenue} = \text{Price per unit} \times \text{quantity sold} Given that each cookie is sold for P25.00P25.00, the total revenue is: TR(x)=25x\text{TR}(x) = 25x

  • Total Cost (TC): This is the sum of the fixed costs and the total variable cost. Total Cost=Fixed Cost+Total Variable Cost\text{Total Cost} = \text{Fixed Cost} + \text{Total Variable Cost} The fixed cost is given as P790.00P790.00. The total variable cost is the cost to make each cookie multiplied by the number of cookies sold, so: Total Variable Cost=15x\text{Total Variable Cost} = 15x Therefore, the total cost is: TC(x)=790+15x\text{TC}(x) = 790 + 15x

  • Profit Function (P(x)): Profit is the difference between total revenue and total cost. P(x)=TR(x)TC(x)P(x) = \text{TR}(x) - \text{TC}(x) Substituting the expressions we have: P(x)=25x(790+15x)P(x) = 25x - (790 + 15x) Simplifying this: P(x)=25x79015x=10x790P(x) = 25x - 790 - 15x = 10x - 790

Part b: Total Profit for 146 Cookies Sold

To find the profit when 146 cookies are sold, substitute x=146x = 146 into the profit function P(x)P(x):

P(146)=10(146)790P(146) = 10(146) - 790 P(146)=1460790=670P(146) = 1460 - 790 = 670

So, the total profit when 146 cookies are sold is P670.00P670.00.

Part c: Break-even Point

The break-even point is when the profit is zero, P(x)=0P(x) = 0. Set the profit function equal to zero and solve for xx:

0=10x7900 = 10x - 790 10x=79010x = 790 x=79010=79x = \frac{790}{10} = 79

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 P250.00P250.00, set P(x)=250P(x) = 250 and solve for xx:

250=10x790250 = 10x - 790 10x=250+790=104010x = 250 + 790 = 1040 x=104010=104x = \frac{1040}{10} = 104

So, 104 cookies need to be sold to gain a profit of P250.00P250.00.


Would you like further details or have any questions?

Here are some related questions:

  1. What is the meaning of the break-even point in business?
  2. How would the profit function change if the selling price increased?
  3. What is the impact of a higher fixed cost on the break-even point?
  4. How could the Bakers' Club maximize its profit?
  5. 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