Problem 20 - Applied Calculus

Question

The following table gives the cost and revenue, in dollars, for different production levels, \(q .\) (a) At approximately what production level is profit maximized? (b) What price is charged per unit for this product? (c) What are the fixed costs of production? \begin{array}{c|c|c|c|c|c|c} \hline q & 0 & 100 & 200 & 300 & 400 & 500 \\ \hline R(q) & 0 & 500 & 1000 & 1500 & 2000 & 2500 \\ \hline C(q) & 700 & 900 & 1000 & 1100 & 1300 & 1900 \\ \hline \end{array}

Step-by-Step Solution

Answer

(a) Profit is maximized at production level 400. (b) The price per unit is $5. (c) Fixed costs are $700.

Step 1: Identify Profit Formula

Profit, denoted as \( P(q) \), is given by the formula \( P(q) = R(q) - C(q) \), where \( R(q) \) is revenue and \( C(q) \) is cost for the production level \( q \).

Step 2: Calculate Profit for Each Production Level

Using the profit formula, calculate the profit for each \( q \):- For \( q = 0 \), \( P(0) = 0 - 700 = -700 \)- For \( q = 100 \), \( P(100) = 500 - 900 = -400 \)- For \( q = 200 \), \( P(200) = 1000 - 1000 = 0 \)- For \( q = 300 \), \( P(300) = 1500 - 1100 = 400 \)- For \( q = 400 \), \( P(400) = 2000 - 1300 = 700 \)- For \( q = 500 \), \( P(500) = 2500 - 1900 = 600 \)

Step 3: Determine Production Level for Maximum Profit

Compare profits calculated in Step 2. The maximum profit occurs at \( q = 400 \) with a profit of 700. Hence, the production level where profit is maximized is \( q = 400 \).

Step 4: Determine Price Charged per Unit

Revenue \( R(q) \) is obtained by multiplying the price per unit \( p \) with the quantity \( q \). For any linear revenue function, the price per unit can be calculated as the rate of change in \( R \) with respect to \( q \). From the table, \( R(q) = 5q \). Thus, the price per unit is \$5.

Step 5: Determine Fixed Costs of Production

Fixed costs \( C_f \) are costs that do not vary with output. When no goods are produced, \( q = 0 \), the total cost \( C(0) = 700 \) represents only the fixed cost. Therefore, the fixed costs are \$700.

Key Concepts

Cost function

A cost function, denoted as \( C(q) \), represents the total cost of producing \( q \) units of a product. It is an essential part of understanding how expenses vary at different levels of output. Cost functions usually account for all production expenses, including materials, labor, utility bills, and more. In some cases, they can also incorporate overhead costs, like facility maintenance and administrative salaries.

In the exercise provided, the function \( C(q) \) is derived from observing the cost related to different production levels. It shows us how total costs increase as the number of units produced increases. Importantly, the cost function can be divided into two parts: fixed costs, which do not change with production level, and variable costs, which do. Recognizing these components is critical when calculating profit and making business decisions.

Revenue function

The revenue function is a key concept in understanding how businesses earn income from selling products or services. It is denoted by \( R(q) \) and represents the total revenue generated when selling \( q \) units. To find this function, one typically multiplies the number of units sold by the price per unit.

For this specific exercise, the revenue function follows a simple linear model: \( R(q) = 5q \). This means for every unit sold, the revenue increases by \$5. Such direct relationships between units sold and revenue simplifies calculations and helps to predict future earnings. This information is crucial for businesses during the planning and decision-making process.

Fixed costs

Fixed costs, or \( C_f \), are expenses that remain constant, regardless of how many units are produced. Examples include rent, salaries, and insurance premiums. Understanding fixed costs helps businesses to determine the break-even point and establish pricing strategies.

In our problem, the fixed cost is evident when no goods are produced, i.e., when \( q = 0 \). At this point, the entire cost \( C(0) = 700 \) is attributed to fixed costs since no variable costs are involved. Knowing the fixed costs allows a company to better manage finances and plan for sustainable growth.

Price per unit

The price per unit is the amount of money a customer pays to buy one unit of a product or service. It plays a crucial role in determining the revenue a business earns and is essential for calculating profit. For revenue function \( R(q) = pq \) where \( p \) is the price per unit, finding \( p \) is necessary to understand the pricing model.

From the exercise, the price per unit is calculated as \\(5, derived from the linear relationship where revenue increases by \\)5 for each additional unit produced and sold. Understanding and setting the right price per unit can influence a business's competitiveness and overall success in the market.