Answer :
Sure, let's break down the given question and provide a detailed step-by-step solution.
### 1.5.1 Explain what is meant by a break-even point in the business context.
In a business context, the break-even point is the point at which total revenue equals total costs. At this point, the business is neither making a profit nor incurring a loss—it is simply "breaking even." This is an important metric for businesses as it helps them understand the minimum amount of sales they need to cover their costs.
### 1.5.2 Determine the values of [tex]\(M\)[/tex] and [tex]\(N\)[/tex].
To determine the values of [tex]\( M \)[/tex] and [tex]\( N \)[/tex], we will use linear interpolation based on the given data in the table.
#### To find [tex]\( M \)[/tex] (Total Cost when 100 dozens sold):
We know the total cost at 0 dozens sold is 2500 and the total cost at 200 dozens sold is 6500. We need to estimate the total cost at 100 dozens sold.
1. Calculate the slope between costs at 0 and 200 dozens sold:
[tex]\[ \text{slope} = \frac{6500 - 2500}{200 - 0} = \frac{4000}{200} = 20 \][/tex]
2. Using the slope, we can interpolate to find the total cost at 100 dozens sold:
[tex]\[ M = 2500 + 20 \times (100 - 0) = 2500 + 2000 = 4500 \][/tex]
So, [tex]\( M = 4500 \)[/tex].
#### To find [tex]\( N \)[/tex] (Total Income when 200 dozens sold):
We know the total income at 100 dozens sold is 4800 and the total income at 300 dozens sold is 14000. We need to estimate the total income at 200 dozens sold.
1. Calculate the slope between incomes at 100 and 300 dozens sold:
[tex]\[ \text{slope} = \frac{14000 - 4800}{300 - 100} = \frac{9200}{200} = 46 \][/tex]
2. Using the slope, we can interpolate to find the total income at 200 dozens sold:
[tex]\[ N = 4800 + 46 \times (200 - 100) = 4800 + 4600 = 9400 \][/tex]
So, [tex]\( N = 9400 \)[/tex].
### 1.5.3 Use the same system of axes provided in ANSWER SHEET 1, to plot a graph to compare the income and total costs with the number of scones sold.
You would plot the following points on a graph:
#### For Total Cost (R):
- (0, 2500)
- (50, 3500)
- (100, 4500)
- (200, 6500)
- (300, 8500)
#### For Total Income (R):
- (0, 0)
- (50, 2400)
- (100, 4800)
- (200, 9400)
- (300, 14000)
### 1.5.4 On the graph:
#### (a) Show the region indicating a profit in Eric's business.
To show the profit region on the graph:
- Identify the area where the total income line is above the total cost line.
- This region indicates where Eric's business is making a profit.
#### (b) How many dozens of scones must he sell to break even?
To determine the break-even point:
- Look for the point on the graph where the total cost and total income lines intersect.
- At this point, total income equals total cost.
From the interpolation and calculations, Eric will break even at approximately 150 dozens sold.
### Conclusion
The break-even analysis offers a clear financial picture, helping Eric understand the requisite sales volume needed to cover costs and start making a profit. These computations and graphs are essential tools for business planning and decision-making.
### 1.5.1 Explain what is meant by a break-even point in the business context.
In a business context, the break-even point is the point at which total revenue equals total costs. At this point, the business is neither making a profit nor incurring a loss—it is simply "breaking even." This is an important metric for businesses as it helps them understand the minimum amount of sales they need to cover their costs.
### 1.5.2 Determine the values of [tex]\(M\)[/tex] and [tex]\(N\)[/tex].
To determine the values of [tex]\( M \)[/tex] and [tex]\( N \)[/tex], we will use linear interpolation based on the given data in the table.
#### To find [tex]\( M \)[/tex] (Total Cost when 100 dozens sold):
We know the total cost at 0 dozens sold is 2500 and the total cost at 200 dozens sold is 6500. We need to estimate the total cost at 100 dozens sold.
1. Calculate the slope between costs at 0 and 200 dozens sold:
[tex]\[ \text{slope} = \frac{6500 - 2500}{200 - 0} = \frac{4000}{200} = 20 \][/tex]
2. Using the slope, we can interpolate to find the total cost at 100 dozens sold:
[tex]\[ M = 2500 + 20 \times (100 - 0) = 2500 + 2000 = 4500 \][/tex]
So, [tex]\( M = 4500 \)[/tex].
#### To find [tex]\( N \)[/tex] (Total Income when 200 dozens sold):
We know the total income at 100 dozens sold is 4800 and the total income at 300 dozens sold is 14000. We need to estimate the total income at 200 dozens sold.
1. Calculate the slope between incomes at 100 and 300 dozens sold:
[tex]\[ \text{slope} = \frac{14000 - 4800}{300 - 100} = \frac{9200}{200} = 46 \][/tex]
2. Using the slope, we can interpolate to find the total income at 200 dozens sold:
[tex]\[ N = 4800 + 46 \times (200 - 100) = 4800 + 4600 = 9400 \][/tex]
So, [tex]\( N = 9400 \)[/tex].
### 1.5.3 Use the same system of axes provided in ANSWER SHEET 1, to plot a graph to compare the income and total costs with the number of scones sold.
You would plot the following points on a graph:
#### For Total Cost (R):
- (0, 2500)
- (50, 3500)
- (100, 4500)
- (200, 6500)
- (300, 8500)
#### For Total Income (R):
- (0, 0)
- (50, 2400)
- (100, 4800)
- (200, 9400)
- (300, 14000)
### 1.5.4 On the graph:
#### (a) Show the region indicating a profit in Eric's business.
To show the profit region on the graph:
- Identify the area where the total income line is above the total cost line.
- This region indicates where Eric's business is making a profit.
#### (b) How many dozens of scones must he sell to break even?
To determine the break-even point:
- Look for the point on the graph where the total cost and total income lines intersect.
- At this point, total income equals total cost.
From the interpolation and calculations, Eric will break even at approximately 150 dozens sold.
### Conclusion
The break-even analysis offers a clear financial picture, helping Eric understand the requisite sales volume needed to cover costs and start making a profit. These computations and graphs are essential tools for business planning and decision-making.
