Problem 90 - College Algebra

Question

The function \(D(p)\) gives the number of items that will be demanded when the price is \(p\). The production \(\operatorname{cost} C(x)\) is the cost of producing \(x\) items. To determine the cost of production when the price is \(\$ 6,\) you would do which of the following? a. Evaluate \(D(C(6))\) b. Evaluate \(C(D(6))\) c. Solve \(D(C(x))=6\). d. Solve \(C(D(p))=6\).

Step-by-Step Solution

Answer

Evaluate \(C(D(6))\).

Step 1: Understanding the Terms

The function \(D(p)\) gives the demand in terms of price \(p\), meaning it indicates how many items are sold at a specific price. The function \(C(x)\) provides the cost to produce \(x\) items.

Step 2: Determine the Demand at Given Price

To find out how many items are demanded at price \(p = 6\), we evaluate \(D(6)\). This step doesn't involve finding the cost yet but just gives us the quantity \(x\), which is needed for subsequent steps.

Step 3: Calculate the Cost of Production

With the quantity \(x\) obtained from \(D(6)\), use the function \(C(x)\) to find the cost of producing these items. Therefore, evaluate \(C(D(6))\).

Key Concepts

Function Evaluation

When dealing with functions, such as demand and production cost functions, it's important to know how to evaluate them correctly. A function like \( D(p) \) or \( C(x) \) links an input to an output. Here, \( D(p) \) represents how many items are demanded at a particular price \( p \), and \( C(x) \) describes the cost of producing \( x \) items. To evaluate these functions means to substitute a given value into the function to calculate the output. For example, if you want to know the demand at a price of $6, you simply input \( p = 6 \) into the demand function, resulting in \( D(6) \). Through this simple step, you can determine how many items are being sought by consumers at that price point. This process of substituting and calculating helps link the theoretical aspect of functions to practical applications, aiding businesses in making informed decisions.

Cost of Production

The concept of the cost of production ties into how much money a company spends to produce a certain number of items. This is represented by the function \( C(x) \), where \( x \) is the number of items. It's a critical factor for businesses as it determines pricing and profitability. Once the demand quantity \( x \) is known from evaluating \( D(6) \), you can then use this quantity to find out how much it costs to produce these items. This involves substituting the quantity into the cost function, evaluating \( C(x) \). By doing so, it's possible to compute the total expense incurred in manufacturing the exact number of items that meet the consumer demand. This function evaluation directly influences production decisions and helps align the company's resources effectively.

Price and Quantity Relationship

Understanding the relationship between price and quantity is fundamental in economics, particularly in studying demand and supply. The demand function \( D(p) \) captures this relationship by illustrating how the number of items consumers demand changes with the price. As price changes, it's natural for demand to respond, often inversely – as price decreases, demand increases and vice versa. In practical terms, to determine how many items are needed at a given price, you input the price into the demand function such as \( D(6) \). The result shows the expected sales volume at this price level. Once this information is known, businesses can decide how many units to produce by looking at the cost of production. By appreciating this dynamic, companies can set optimum prices that maximize both sales and profits, creating an equilibrium that satisfies consumer needs and company goals.