Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A photocopying shop has a xed cost of operation of $$\$ 4000$$ per month. In addition, it costs them $$\$ 0.01$$ per page they copy. They charge customers $$\$ 0.07$$ per page. (a) Write a formula for \(R(x)\), the shop s monthly revenue from making \(x\) copies. (b) Write a formula for \(C(x)\), the shop s monthly costs from making \(x\) copies. (c) Write a formula for \(P(x)\), the shop s monthly pro t (or loss if negative) from making \(x\) copies. Pro \(\mathrm{t}\) is computed by subtracting total costs from the total revenue. (d) How many copies must they make per month in order to break even? Breaking even means that the pro \(t\) is zero; the total costs and total revenue are equal. (e) Sketch \(C(x), R(x)\), and \(P(x)\) on the same set of axes and label the break-even point. (f) Find a formula for \(A(x)\), the shop s average cost per copy. (g) Make a table of \(A(x)\) for \(x=0,1,10,100,1000,10000\). (h) Sketch a graph of \(A(x)\).

Step-by-Step Solution

Answer

The photocopying shop's monthly revenue, cost, and profit equations are \(R(x) = 0.07x\), \(C(x) = 4000 + 0.01x\), and \(P(x) = 0.06x - 4000\), respectively. They need to make about 66667 copies to break even. The average cost per copy, \(A(x)\), is \((4000 + 0.01x) / x\). The values of \(A(x)\) for different x-values can be found by substitution into the equation.

Step 1: Write the Revenue Formula

Revenue, \(R(x)\), is the amount the photocopying shop charges customers per page multiplied by the number of copies, \(x\). So it is \(R(x) = 0.07x\).

Step 2: Write the Cost Formula

The monthly costs, \(C(x)\), consist of the fixed cost, $4000, plus the variable cost of copying each page. So it is \(C(x) = 4000 + 0.01x\).

Step 3: Write the Profit Formula

Profit, \(P(x)\), is calculated by subtracting total costs from total revenue. So it is \(P(x) = R(x) - C(x) = 0.07x - (4000 + 0.01x) = 0.06x - 4000\).

Step 4: Calculate the Breakeven Point

At the breakeven point, profit (P(x)) is zero. So, set the equation \(0.06x - 4000 = 0\) and solve for \(x\). This gives \(x = 66666.67\). So, they need to make approximately 66667 copies to break even.

Step 5: Create the Graph

Plot the revenue, cost, and profit equations on the same graph. The intersection point of the revenue and cost lines indicates the breakeven point.

Step 6: Write the Average Cost Formula

Average cost, \(A(x)\), is the total cost divided by the number of copies. So it is \(A(x) = C(x)/x = (4000 + 0.01x) / x \).

Step 7: Construct the Table of A(x)

Create a table with different x-values (0,1,10,100,1000,10000), calculate the corresponding average cost using \(A(x)\) and present them in the table.

Step 8: Sketch the Graph of A(x)

Plot the graph of \(A(x)\) using the table created in Step 7. Observe how the average cost changes with the number of copies.

Key Concepts

Revenue Calculation

Understanding how to calculate revenue is crucial for any business. In the case of the photocopying shop, revenue, denoted as \( R(x) \), represents the total money received from customers for the copies made. To find this, you simply multiply the number of copies \( x \) by the price charged per copy, which is \$0.07. The formula for revenue thus becomes:

• Revenue Formula: \( R(x) = 0.07x \)

This straightforward calculation helps businesses predict income based on sales and pricing.

Profit Analysis

Profit analysis allows businesses to understand their financial health by assessing earnings after expenses. Profit \( P(x) \) is derived by subtracting total costs from total revenue. For the photocopying shop, using the revenue \( R(x) = 0.07x \) and cost \( C(x) = 4000 + 0.01x \) formulas, the profit formula becomes:

• Profit Formula: \( P(x) = 0.06x - 4000 \)

This equation reflects the importance of increasing production to cover fixed costs and achieve profitability.

Break-Even Point

The break-even point is where a business neither makes a profit nor incurs a loss. It's a vital metric, indicating when total costs are equal to total revenue. For the shop, this is where \( P(x) = 0 \), leading to:

• Break-even Equation: \( 0.06x - 4000 = 0 \)
• Solution: \( x = 66667 \)

This means approximately 66,667 copies need to be made to break even, highlighting how cost management impacts business operations.

Average Cost Formula

The average cost formula helps in understanding the cost efficiency per unit produced. For the shop, average cost \( A(x) \) is calculated by dividing total costs \( C(x) = 4000 + 0.01x \) by the number of copies \( x \):

• Average Cost Formula: \( A(x) = \frac{4000 + 0.01x}{x} \)

As production increases, the average cost per copy decreases, emphasizing economies of scale. Calculating average cost assists in pricing strategy and financial planning.