Answer :
Sure! Let's break down the questions step-by-step.
### 1.5.1 Explain what is meant by a break-even point in the business context.
The break-even point in business is the point at which total revenues (income) equal total costs. At this point, a business neither makes a profit nor incurs a loss. It is essentially the level of sales at which all the costs of producing and selling a product or service are covered.
### 1.5.2 Determine the values of [tex]\(M\)[/tex] and [tex]\(N\)[/tex].
#### Finding [tex]\(M\)[/tex] (when dozens sold = 100):
To find [tex]\(M\)[/tex], we can use linear interpolation between the given total costs at 50 and 200 dozens sold:
[tex]\[ \text{Slope (m)} = \frac{\text{Total Cost at 200 dozens} - \text{Total Cost at 50 dozens}}{200 - 50} = \frac{6500 - 3500}{150} = \frac{3000}{150} = 20 \][/tex]
Then, using the point-slope form of the linear equation:
[tex]\[ \text{Total Cost at 100 dozens} = \text{Total Cost at 50 dozens} + \text{Slope} \times (\text{Number of dozens at 100} - \text{Number of dozens at 50}) \][/tex]
[tex]\[ M = 3500 + 20 \times (100 - 50) \][/tex]
[tex]\[ M = 3500 + 20 \times 50 \][/tex]
[tex]\[ M = 3500 + 1000 \][/tex]
[tex]\[ M = 4500 \][/tex]
So, [tex]\(M = 4500\)[/tex].
#### Finding [tex]\(N\)[/tex] (when dozens sold = 200):
To find [tex]\(N\)[/tex], we use linear interpolation between the given total incomes at 100 and 300 dozens sold:
[tex]\[ \text{Slope (m)} = \frac{\text{Total Income at 300 dozens} - \text{Total Income at 100 dozens}}{300 - 100} = \frac{1400 - 4800}{200} = \frac{-3400}{200} = -17 \][/tex]
Then, using the point-slope form of the linear equation:
[tex]\[ \text{Total Income at 200 dozens} = \text{Total Income at 100 dozens} + \text{Slope} \times (\text{Number of dozens at 200} - \text{Number of dozens at 100}) \][/tex]
[tex]\[ N = 4800 + (-17) \times (200 - 100) \][/tex]
[tex]\[ N = 4800 - 17 \times 100 \][/tex]
[tex]\[ N = 4800 - 1700 \][/tex]
[tex]\[ N = 3100 \][/tex]
So, [tex]\(N = 3100\)[/tex].
### 1.5.3 Plot a Graph to Compare Income and Total Costs
While I cannot physically plot a graph here, I can describe how to do this on your provided ANSWER SHEET 1.
1. Label the X-axis as "Number of Dozens Sold".
2. Label the Y-axis as "Amount (R)".
3. Plot the total cost points at (0, 2500), (50, 3500), (100, 4500), (200, 6500), and (300, 8500).
4. Plot the total income points at (0, 0), (50, 2400), (100, 4800), (200, 3100), and (300, 1400).
5. Draw a straight line connecting the total cost points and another line connecting the total income points.
### 1.5.4 Use the Graph to Answer the Following:
#### (a) Show the Region of Profit:
The region of profit is where the total income line is above the total cost line. Shade this area on your graph.
#### (b) Determining the Break-Even Point:
The break-even point is where the total income line intersects the total cost line. By inspection from your plotted graph, this should visually show where Eric must sell dozens of scones to cover all his costs.
In summary:
- The value of [tex]\(M\)[/tex] is 4500.
- The value of [tex]\(N\)[/tex] is 3100.
- The region of profit is where the income line is above the cost line.
- The break-even point occurs at the intersection of the income and cost lines, which you can determine from your graph. To find the exact break-even number of dozens, look for the x-value at which these y-values are equal.
From the interpolation and concept, one might deduce that the break-even does show when income equals cost, closer to the calculation point. For mathematical purposes, we must rely on solving [tex]\(y_{income} = y_{cost}\)[/tex].
### 1.5.1 Explain what is meant by a break-even point in the business context.
The break-even point in business is the point at which total revenues (income) equal total costs. At this point, a business neither makes a profit nor incurs a loss. It is essentially the level of sales at which all the costs of producing and selling a product or service are covered.
### 1.5.2 Determine the values of [tex]\(M\)[/tex] and [tex]\(N\)[/tex].
#### Finding [tex]\(M\)[/tex] (when dozens sold = 100):
To find [tex]\(M\)[/tex], we can use linear interpolation between the given total costs at 50 and 200 dozens sold:
[tex]\[ \text{Slope (m)} = \frac{\text{Total Cost at 200 dozens} - \text{Total Cost at 50 dozens}}{200 - 50} = \frac{6500 - 3500}{150} = \frac{3000}{150} = 20 \][/tex]
Then, using the point-slope form of the linear equation:
[tex]\[ \text{Total Cost at 100 dozens} = \text{Total Cost at 50 dozens} + \text{Slope} \times (\text{Number of dozens at 100} - \text{Number of dozens at 50}) \][/tex]
[tex]\[ M = 3500 + 20 \times (100 - 50) \][/tex]
[tex]\[ M = 3500 + 20 \times 50 \][/tex]
[tex]\[ M = 3500 + 1000 \][/tex]
[tex]\[ M = 4500 \][/tex]
So, [tex]\(M = 4500\)[/tex].
#### Finding [tex]\(N\)[/tex] (when dozens sold = 200):
To find [tex]\(N\)[/tex], we use linear interpolation between the given total incomes at 100 and 300 dozens sold:
[tex]\[ \text{Slope (m)} = \frac{\text{Total Income at 300 dozens} - \text{Total Income at 100 dozens}}{300 - 100} = \frac{1400 - 4800}{200} = \frac{-3400}{200} = -17 \][/tex]
Then, using the point-slope form of the linear equation:
[tex]\[ \text{Total Income at 200 dozens} = \text{Total Income at 100 dozens} + \text{Slope} \times (\text{Number of dozens at 200} - \text{Number of dozens at 100}) \][/tex]
[tex]\[ N = 4800 + (-17) \times (200 - 100) \][/tex]
[tex]\[ N = 4800 - 17 \times 100 \][/tex]
[tex]\[ N = 4800 - 1700 \][/tex]
[tex]\[ N = 3100 \][/tex]
So, [tex]\(N = 3100\)[/tex].
### 1.5.3 Plot a Graph to Compare Income and Total Costs
While I cannot physically plot a graph here, I can describe how to do this on your provided ANSWER SHEET 1.
1. Label the X-axis as "Number of Dozens Sold".
2. Label the Y-axis as "Amount (R)".
3. Plot the total cost points at (0, 2500), (50, 3500), (100, 4500), (200, 6500), and (300, 8500).
4. Plot the total income points at (0, 0), (50, 2400), (100, 4800), (200, 3100), and (300, 1400).
5. Draw a straight line connecting the total cost points and another line connecting the total income points.
### 1.5.4 Use the Graph to Answer the Following:
#### (a) Show the Region of Profit:
The region of profit is where the total income line is above the total cost line. Shade this area on your graph.
#### (b) Determining the Break-Even Point:
The break-even point is where the total income line intersects the total cost line. By inspection from your plotted graph, this should visually show where Eric must sell dozens of scones to cover all his costs.
In summary:
- The value of [tex]\(M\)[/tex] is 4500.
- The value of [tex]\(N\)[/tex] is 3100.
- The region of profit is where the income line is above the cost line.
- The break-even point occurs at the intersection of the income and cost lines, which you can determine from your graph. To find the exact break-even number of dozens, look for the x-value at which these y-values are equal.
From the interpolation and concept, one might deduce that the break-even does show when income equals cost, closer to the calculation point. For mathematical purposes, we must rely on solving [tex]\(y_{income} = y_{cost}\)[/tex].
