A Company expects fixed
cost of $36,000. Margin is to
be 52 percent of retail and
variable cost in addition to
cost of goods is estimated
at $0.07 per dollar of sales.
a) Find the revenue, cost and
profit functions using s for
sales volume.
b) Find the break-even point.
c) What will net profit before
taxes be on sales of
$75,000?
d) Draw the break-even chart.

### Part (a): Finding the Revenue, Cost, and Profit Functions

Let:
- $ s $ = Sales volume in dollars.

#### **Revenue Function ($ R(s) $)**:
The revenue is the amount the company earns from sales. Given that the margin is 52% of the retail price, this means 52% of the sales volume $ s $ will be considered as revenue:

\[
R(s) = s
\]

#### **Cost Function ($ C(s) $)**:
The cost function consists of the fixed costs and variable costs. We are given:
- Fixed cost = $36,000
- Variable cost = $0.07 per dollar of sales, which represents additional variable costs.

The total cost function can be expressed as:

\[
C(s) = 36000 + 0.07s
\]

#### **Profit Function ($ P(s) $)**:
The profit function is obtained by subtracting the total cost from the revenue:

\[
P(s) = R(s) - C(s)
\]

Substituting the expressions for $ R(s) $ and $ C(s) $:

\[
P(s) = s - (36000 + 0.07s)
\]

\[
P(s) = s - 36000 - 0.07s
\]

\[
P(s) = 0.93s - 36000
\]

---

### Part (b): Break-even Point

At the break-even point, the revenue equals the total cost, i.e., $ R(s) = C(s) $. 

Thus:

\[
s = 36000 + 0.07s
\]

Rearranging the terms to isolate $ s $:

\[
s - 0.07s = 36000
\]

\[
0.93s = 36000
\]

\[
s = \frac{36000}{0.93}
\]

\[
s = 38709.68
\]

So, the break-even point is approximately **$38,710**.

---

### Part (c): Net Profit Before Taxes on Sales of $75,000

To find the net profit before taxes when sales are $75,000, we use the profit function $ P(s) $:

\[
P(75000) = 0.93(75000) - 36000
\]

\[
P(75000) = 69750 - 36000
\]

\[
P(75000) = 33750
\]

Therefore, the net profit before taxes is **$33,750**.

---

### Part (d): Drawing the Break-even Chart

1. **Axes**: Let the x-axis represent the sales volume $ s $ in dollars, and the y-axis represent the costs/revenue in dollars.
2. **Revenue Line**: Plot the revenue line $ R(s) = s $, which is a straight line starting from the origin (0,0) with a slope of 1.
3. **Cost Line**: Plot the cost line $ C(s) = 36000 + 0.07s $, which starts at (0, 36000) with a slope of 0.07.
4. **Break-even Point**: The point where the revenue line and cost line intersect at approximately $ s = 38710 $.

Since I can't directly draw the chart here, you can visualize it as follows:
- The revenue line is steeper than the cost line.
- The intersection at around $ s = 38710 $ represents the break-even point.

Would you like any more details or do you have any questions?

---

### Relative Questions:
1. How does a 10% increase in fixed costs affect the break-even point?
2. What would be the new profit function if the variable cost increased to $0.10 per dollar of sales?
3. How would a reduction in the margin percentage impact the revenue function?
4. What happens to the break-even point if fixed costs decrease by $5,000?
5. How do changes in sales volume affect profit if fixed costs remain unchanged?

**Tip**: Always analyze the break-even point to determine how changes in costs or sales pricing impact profitability. This helps in making informed decisions about pricing and cost controls.

A Company expects fixed cost of $36,000. Margin is to be 52 percent of retail and variable cost in addition to cost of goods is estimated at $0.07 per dollar of sales. a) Find the revenue, cost and profit functions using s for sales volume. b) Find the break-even point. c) What will net profit before taxes be on sales of $75,000? d) Draw the break-even chart.

Learn how to analyze a company's break-even point with a detailed step-by-step solution. This problem covers key concepts such as revenue, cost, and profit functions, as well as finding the break-even point. Suitable for students in Grades 11-12.

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