4.1 What type of relationship is illustrated by inis graph of the 'Income recelved' based on the 'Number of cups of caffeg sold'? Glve a reason for your answer. 4.2 Datermine the formula for the Income recelyed. 4.3 How much income would be recolved if 3 cups of caffee were sold? 4.4 How many cups of coffee need to be sold to recetive an income of R3M?

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Certainly! Let's address each of your questions step-by-step. Since the graph isn't provided, I'll base my answers on common relationships between income and the number of items sold, assuming a linear relationship unless specified otherwise.

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### **4.1. Type of Relationship Illustrated by the Graph**

**Answer:**
The relationship between **'Income Received'** and the **'Number of Cups of Coffee Sold'** is a **linear (direct) relationship**.

**Reason:**
In a linear relationship, the income increases at a constant rate as the number of cups sold increases. This means that each additional cup sold contributes the same amount to the total income, indicating a proportional and straight-line relationship on the graph.

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### **4.2. Formula for the Income Received**

**Answer:**
Assuming a linear relationship, the formula to calculate the **Income Received (I)** based on the **Number of Cups Sold (c)** is:

$$ \text{Income (I)} = \text{Price per Cup (p)} \times \text{Number of Cups Sold (c)} $$

**Mathematically:**
$$ I = p \times c $$

- **I** = Income Received
- **p** = Price per cup of coffee
- **c** = Number of cups sold

*If there are fixed costs involved (costs that do not change with the number of cups sold), the formula would incorporate those as well:*

$$ I = (\text{Price per Cup} \times \text{Number of Cups Sold}) - \text{Fixed Costs} $$

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### **4.3. Income Received if 3 Cups of Coffee are Sold**

**Answer:**
Using the formula derived in **4.2**, the income received from selling 3 cups of coffee is:

$$ I = p \times 3 $$

**Example Calculation:**
If the price per cup (*p*) is **R10**, then:

$$ I = 10 \times 3 = R30 $$

*Therefore, selling 3 cups of coffee would generate **R30** in income.*

*Without the specific price per cup, the income can be expressed as:*

$$ I = 3p $$

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### **4.4. Number of Cups Needed to be Sold to Receive an Income of R3,000,000**

**Answer:**
To determine the number of cups (*c*) needed to achieve an income (*I*) of **R3,000,000**, rearrange the formula from **4.2**:

$$ c = \frac{I}{p} $$

**Plugging in the values:**

$$ c = \frac{3,000,000}{p} $$

**Example Calculation:**
If the price per cup (*p*) is **R10**, then:

$$ c = \frac{3,000,000}{10} = 300,000 \text{ cups} $$

*Therefore, you would need to sell **300,000 cups** of coffee to receive an income of **R3,000,000**.*

*If the price per cup is different, simply replace **R10** with the actual price to find the required number of cups.*

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**Summary:**
- The relationship is linear, meaning income increases proportionally with the number of cups sold.
- The basic formula is $$ I = p \times c $$.
- Specific income or required sales can be calculated by plugging in the known values for price per cup and desired income.

Feel free to provide more details or specific numbers if you need more precise calculations!
**4.1. Relationship Type:** The relationship is linear, showing that income increases at a constant rate with each cup sold. **4.2. Income Formula:** Income = Price per Cup × Number of Cups Sold **4.3. Income for 3 Cups:** If each cup is R10, income = 3 × R10 = R30 **4.4. Cups Needed for R3,000,000:** Number of Cups = R3,000,000 ÷ Price per Cup. If each cup is R10, then 300,000 cups are needed.

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