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 $C(s)$:

$P(s) = s - (36000 + 0.07s)$

$P(s) = s - 36000 - 0.07s$

$P(s) = 0.93s - 36000$

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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.

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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.

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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?

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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.