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It's general mathematics of grade 11 humss

The image shows a problem related to profit calculation in a bakery business. Let's break down the problem and solve each part.

### Problem Summary:
- **Bakery Name**: Baker's Nook
- **Item Sold**: Doughnuts
- **Selling Price per Doughnut**: ₱20.00
- **Cost of Making per Doughnut**: ₱10.00
- **Daily Operating Expense**: ₱1,000.00

### Questions:
a. **Find the profit function \( P(x) \), where \( x \) represents the number of doughnuts sold.**
b. **If 100 doughnuts were sold, what is the total profit?**
c. **How many doughnuts should be made and sold to break even?**
d. **How many doughnuts should be sold to gain a profit of ₱4,000?**

### Solutions:

#### Part a: Profit Function \( P(x) \)
The profit function \( P(x) \) is calculated as the total revenue minus the total cost.

- **Revenue** from selling \( x \) doughnuts: \( R(x) = 20x \) 
- **Cost** of making \( x \) doughnuts: \( C(x) = 10x + 1000 \) 
  - (₱10.00 per doughnut + ₱1,000.00 fixed cost)

Thus, the **profit function** is:
\[
P(x) = R(x) - C(x) = 20x - (10x + 1000) = 10x - 1000
\]

#### Part b: Total Profit when 100 Doughnuts are Sold
Substitute \( x = 100 \) into the profit function:
\[
P(100) = 10(100) - 1000 = 1000 - 1000 = ₱0
\]
So, the profit is ₱0 when 100 doughnuts are sold.

#### Part c: Break-even Point
To find the break-even point, set the profit function equal to zero and solve for \( x \):
\[
P(x) = 10x - 1000 = 0
\]
\[
10x = 1000
\]
\[
x = 100
\]
So, the break-even point is at 100 doughnuts. This means the bakery needs to sell 100 doughnuts to cover all costs and achieve zero profit.

#### Part d: Doughnuts Needed to Gain a Profit of ₱4,000
Set \( P(x) = 4000 \) and solve for \( x \):
\[
10x - 1000 = 4000
\]
\[
10x = 5000
\]
\[
x = 500
\]
So, 500 doughnuts need to be sold to gain a profit of ₱4,000.

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### Do you want details or have any questions? Here are some related questions to consider:

1. How is the break-even point calculated in general?
2. What is the significance of the profit function in business?
3. Can the profit function be used to estimate losses?
4. How does an increase in operating expenses affect the profit function?
5. What would happen to the break-even point if the cost per doughnut increased?
6. How would changes in the selling price affect the number of doughnuts needed to achieve a specific profit?
7. Could the concept of marginal cost be applied to this problem? How?
8. What are some strategies a business could use to lower the break-even point?

**Tip:** When setting up a profit function, always ensure that all variable and fixed costs are accounted for to accurately model the business scenario.

Explore a detailed solution to a grade 11 HUMSS general mathematics problem on profit calculation in a bakery business. Learn how to derive the profit function, calculate total profit, determine the break-even point, and analyze the impact of costs and sales on profitability.

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