Questions: Identify the different types of equations (functions) which are often used in business models. Cost equations or functions relate the quantity, x, and the price, p. Demand equations or functions can be represented by the product, x p, where x is the quantity and p is the price. Profit equations or functions can be represented as the difference, R(x)-C(x).

Solution Steps

To identify the different types of equations (functions) often used in business models, we can categorize them as follows:

1. Cost Equations: These relate the quantity \( x \) and the price \( p \). Typically, cost functions can be linear or quadratic, depending on the complexity of the cost structure.
2. Demand Equations: These can be represented by the product \( x \cdot p \), where \( x \) is the quantity and \( p \) is the price. Demand functions often show how the quantity demanded varies with price.
3. Profit Equations: These are represented as the difference between revenue \( R(x) \) and cost \( C(x) \). The revenue function \( R(x) \) is usually the product of quantity and price, while the cost function \( C(x) \) can vary.

The cost function is given by:

\[ C(x) = \text{fixed cost} + \text{variable cost per unit} \cdot x \]

Substituting the values:

\[ C(100) = 500 + 5 \cdot 100 = 500 + 500 = 1000 \]

The demand function is represented as:

\[ D(x) = x \cdot p \]

Substituting the values:

\[ D(100) = 100 \cdot 20 = 2000 \]

The profit function is defined as:

\[ P(x) = R(x) - C(x) \]

Where the revenue function \( R(x) \) is:

\[ R(x) = D(x) = x \cdot p \]

Thus, substituting the values:

\[ P(100) = D(100) - C(100) = 2000 - 1000 = 1000 \]

Final Answer