Problem 20 - Applied Calculus

Question

A company producing jigsaw puzzles has fixed costs of 6000 dollars and variable costs of 2 dollars per puzzle. The company sells the puzzles for 5 dollars each. (a) Find formulas for the cost function, the revenue function, and the profit function. (b) Sketch a graph of \(R(q)\) and \(C(q)\) on the same axes. What is the break- even point, \(q_{0},\) for the company?

Step-by-Step Solution

Answer

Cost function: \(C(q) = 6000 + 2q\), revenue function: \(R(q) = 5q\), profit: \(P(q) = 3q - 6000\). Break-even at 2000 puzzles.

Step 1: Define the Cost Function

The cost function, denoted by \(C(q)\), is the sum of fixed costs and variable costs. The fixed cost is \(6000\) dollars, and the variable cost is \(2\) dollars per puzzle. Therefore, the cost function is \(C(q) = 6000 + 2q\).

Step 2: Define the Revenue Function

The revenue function, denoted by \(R(q)\), is the total income from selling \(q\) puzzles at \(5\) dollars each. Thus, the revenue function is \(R(q) = 5q\).

Step 3: Define the Profit Function

The profit function \(P(q)\) is the revenue function minus the cost function. Substituting the expressions we have: \[P(q) = R(q) - C(q) = 5q - (6000 + 2q) = 3q - 6000\].

Step 4: Find the Break-even Point

To find the break-even point \(q_0\), we set the profit function equal to zero: \[3q - 6000 = 0\]. Solve for \(q\) to find that \(q_0 = \frac{6000}{3} = 2000\).

Step 5: Sketch Revenue and Cost Functions

On a graph, plot both functions \(R(q) = 5q\) and \(C(q) = 6000 + 2q\). The break-even point is where these two lines intersect, which is at \(q = 2000\) puzzles.

Key Concepts

Cost Function

The cost function is an essential component of any business model. It allows us to calculate the total costs incurred when producing a certain number of units, such as puzzles in this case. The two main components of the cost function are:

• Fixed Costs: These are the costs that must be paid regardless of how many units are produced. For the jigsaw puzzle company, this amount is 6000 dollars.
• Variable Costs: These are the costs associated with producing each additional unit. Here, it is 2 dollars per puzzle.

The cost function, represented as \(C(q)\), combines these two types of costs, giving us the formula:\[ C(q) = 6000 + 2q \]Where \(q\) represents the number of puzzles produced. Thus, this formula helps businesses understand how costs increase with production.

Revenue Function

The revenue function determines the total income a company makes from selling its products. It is crucial for assessing sales performance and planning future business strategies. In our puzzle company example:

• Each puzzle is sold for 5 dollars.
• The revenue function, \(R(q)\), thus becomes:\[ R(q) = 5q \]

This equation shows that revenue is directly proportional to the number of units sold. It's a straightforward way to see how increasing sales impacts income.
Understanding the revenue function is vital, as it lays the foundation for calculating profit and strategically forecasting how changes in price or quantity may affect total earnings.

Profit Function

The profit function is critically important, as it defines the actual earnings after all costs are subtracted from total revenue. It is represented by \(P(q)\).

• The formula for the profit function is derived by subtracting the cost function \(C(q)\) from the revenue function \(R(q)\):

\[ P(q) = R(q) - C(q) \]Plugging in our existing formulas, we get:\[ P(q) = 5q - (6000 + 2q) = 3q - 6000 \]This simplified formula shows how profit varies with changes in the number of items sold. By understanding this function, companies can evaluate how production and sales affect overall profitability and identify opportunities to boost earnings.

Break-even Point

The break-even point is a vital business concept, marking the threshold where total revenues equal total costs. It means this is the point where there is no profit or loss.

• To find the break-even point \(q_0\), set the profit function to zero:

\[ 3q - 6000 = 0 \]Solving for \(q\), we find:\[ q_0 = \frac{6000}{3} = 2000 \]This calculation implies that the company must sell 2000 puzzles to cover all costs. Anything sold beyond this point contributes to profit.
Understanding the break-even point is essential for businesses to make informed decisions about pricing, cost management, and sales targets.