Problem 15 - Applied Calculus

Question

Production costs for manufacturing running shoes consist of a fixed overhead of $$\$ 650,000$$ plus variable costs of $$\$ 20$$ per pair of shoes. Each pair of shoes sells for $$\$ 70$$. (a) Find the total cost, \(C(q)\), the total revenue, \(R(q)\), and the total profit, \(\pi(q)\), as a function of the number of pairs of shoes produced, \(q .\) (b) Find the marginal cost, marginal revenue, and marginal profit. (c) How many pairs of shoes must be produced and sold for the company to make a profit?

Step-by-Step Solution

Answer

The company must produce and sell more than 13,000 pairs of shoes to make a profit.

Step 1: Define the Total Cost Function

The total cost, \(C(q)\), consists of fixed and variable costs. The fixed overhead is \\(650,000 and the variable cost is \\)20 per pair of shoes. Therefore, the total cost function is:\[ C(q) = 650,000 + 20q \]

Step 2: Define the Total Revenue Function

The selling price per pair of shoes is \$70. Therefore, the total revenue, \(R(q)\), from selling \(q\) pairs of shoes is:\[ R(q) = 70q \]

Step 3: Define the Total Profit Function

The total profit, \(\pi(q)\), is the total revenue minus the total cost. So, we have:\[ \pi(q) = R(q) - C(q) = 70q - (650,000 + 20q) \]Simplifying, the profit function is:\[ \pi(q) = 50q - 650,000 \]

Step 4: Find the Marginal Cost

The marginal cost is the derivative of the total cost function with respect to \(q\). Since the total cost function is \(C(q) = 650,000 + 20q\), its derivative is:\[ MC = \frac{d}{dq}(650,000 + 20q) = 20 \]

Step 5: Find the Marginal Revenue

The marginal revenue is the derivative of the total revenue function with respect to \(q\). Since the total revenue function is \(R(q) = 70q\), its derivative is:\[ MR = \frac{d}{dq}(70q) = 70 \]

Step 6: Find the Marginal Profit

The marginal profit is the derivative of the profit function. Since \(\pi(q) = 50q - 650,000\), its derivative is:\[ M\pi = \frac{d}{dq}(50q - 650,000) = 50 \]

Step 7: Determine Break-Even Point for Profit

The company makes a profit when the total profit, \(\pi(q)\), is greater than zero. Solve \(50q - 650,000 > 0\):\[ 50q > 650,000 \]\[ q > \frac{650,000}{50} = 13,000 \]Therefore, the company must produce and sell more than 13,000 pairs of shoes to make a profit.

Key Concepts

Understanding Cost Functions

A crucial part of profit analysis is understanding cost functions. It involves determining how much it costs to produce goods, which includes both fixed and variable costs.
Fixed costs are expenses that do not change with the level of output produced. In this shoe manufacturing example, the fixed overhead is \(\\)650,000\(. This amount stays the same regardless of how many pairs of shoes are produced, whether it is one pair or one million pairs.
Variable costs, on the other hand, depend directly on the number of goods produced. Here, the cost to produce one pair of shoes is \)\\(20\). Thus, as more shoes are produced, the total variable costs increase.
Together, these give us the total cost function. It is expressed mathematically as:

• \(C(q) = 650,000 + 20q\)

where \(C(q)\) is the total cost based on the number of shoe pairs \(q\).
Recognizing the components of this function helps in planning production and budgeting effectively.

Exploring Marginal Analysis

Marginal analysis helps in understanding the effect of producing one additional unit on costs and revenues. Here, we delve into the marginal cost, marginal revenue, and marginal profit.
The marginal cost (MC) is the cost of producing one more pair of shoes. It's found by differentiating the total cost function with respect to \(q\), resulting in:

• \(MC = 20\)

This means it costs an additional \(\\)20\( to produce an extra pair of shoes.
Marginal revenue (MR) is the revenue from selling one more pair. Calculating it means differentiating the total revenue function, \(R(q) = 70q\), giving:

• \(MR = 70\)

Each additional pair sold brings in \)\\(70\) in revenue.
Finally, marginal profit (M\( \pi \)) is the additional profit from one more unit sold, determined by the derivative of the profit function, \( \pi(q) = 50q - 650,000\):

• M\( \pi = 50 \)

This shows that each extra pair yields a profit of \(\\)50$.
Marginal analysis is key to optimizing production and maximizing company profits.

Finding the Break-Even Point

The break-even point is where the company neither makes a profit nor incurs a loss. At this point, total revenue equals total cost, and net profit is zero.
To find this point for our shoe company, we set the profit function \( \pi(q) = 50q - 650,000 \) equal to zero and solve for \(q\):
Steps:

• Set \(50q - 650,000 = 0\)
• Rearrange: \(50q = 650,000\)
• Solve for \(q\): \(q = \frac{650,000}{50} \)
• \(q = 13,000\)

Thus, the company must produce and sell more than 13,000 pairs of shoes to achieve a profit.
Understanding the break-even point helps businesses determine the minimum output level required to cover all expenses, aiding in strategic planning and decision-making.