Problem 11 - Applied Calculus

Question

Give the cost, revenue, and profit functions. An online seller of T-shirts pays $$ 500$ to start up the website and $$ 6 per T-shirt, then sells the T-shirts for $$ 12 each.

Step-by-Step Solution

Answer

Cost: \(C(x) = 500 + 6x\), Revenue: \(R(x) = 12x\), Profit: \(P(x) = 6x - 500\).

Step 1: Understand the Total Cost

The total cost for the online seller includes a fixed cost of \(500 and a variable cost of \)6 per T-shirt. If the number of T-shirts sold is represented as \(x\), then the total cost function \(C(x)\) is given by adding these components: \(C(x) = 500 + 6x\).

Step 2: Understand the Revenue

The revenue is the total money earned from selling the T-shirts. Since each T-shirt is sold for $12, the revenue function \(R(x)\) is the price per T-shirt times the number of T-shirts sold: \(R(x) = 12x\).

Step 3: Calculate the Profit Function

Profit is the difference between revenue and total cost. The profit function \(P(x)\) can be found using the formula \(P(x) = R(x) - C(x)\). Substituting the expressions we have: \(P(x) = 12x - (500 + 6x)\). Simplifying this gives the profit function: \(P(x) = 6x - 500\).

Key Concepts

Cost Function

The cost function is a key concept when analyzing how businesses manage their expenses. For the online seller of T-shirts, the cost function includes two parts:

• Fixed Costs: These are the expenses that don't change regardless of how many T-shirts are sold. In this case, the fixed cost is \(500, which is spent on starting up the website.
• Variable Costs: These depend on the number of items produced or sold. Here, each T-shirt costs an additional \)6 to produce.

The formula to calculate the total cost function, denoted as \(C(x)\), is simply the sum of the fixed cost and the variable cost per unit times the quantity, \(x\): \[C(x) = 500 + 6x\] This formula allows the seller to determine the total expenses at any production level.

Revenue Function

The revenue function represents the total income gained from selling products. For the online T-shirt seller, each T-shirt fetches $12. Hence, the revenue depends on how many T-shirts are sold.To express this as a function, let \(R(x)\) be the revenue function, where \(x\) is the number of T-shirts sold. The formula is:\[R(x) = 12x\]This formula helps to calculate the total earnings before considering any costs, by multiplying the price per T-shirt by the number of T-shirts sold.

Variable Cost

Variable costs change with the level of production or sales. For the T-shirt seller, this cost varies depending on how many T-shirts are sold.

• T-Shirt Cost: The variable cost per T-shirt is \(6. This means, for every T-shirt made or sold, the cost increases by \)6.

To calculate the total variable costs for \(x\) T-shirts, you simply multiply the per-unit cost by the number of units, providing flexibility in managing operations.Understanding variable costs is crucial for pricing strategies and profit planning, as they directly impact the cost function component \(6x\) in \(C(x) = 500 + 6x\).

Fixed Cost

Fixed costs are expenses that do not vary with production or sales volume. No matter how many or few T-shirts are sold, these costs remain constant.

• Website Setup: For the online T-shirt business, the fixed cost is $500, which is the initial investment required for setting up the e-commerce platform.

Dealing with fixed costs effectively is vital for startups to manage initial losses and calculate break-even points, as they are a crucial part of the total cost equation \(C(x) = 500 + 6x\). Understanding where your fixed costs fall helps in planning and forecasting for a business's financial health.