Question

The demand function for a limited edition comic book is given by \(p=3000\left(1-\frac{5}{5+e^{-0.015 x}}\right)\) (a) Find the price \(p\) for a demand of \(x=75\) units. (b) Find the price \(p\) for a demand of \(x=200\) units. (c) Use a graphing utility to graph the demand function. (d) Use the graph from part (c) to approximate the demand when the price is \(\$ 100\).

Short answer

The exact prices for demand of 75 and 200 units would be determined by evaluating the function. For the approximation, it would be the \( x \) value(s) obtained at the intersection of the graph of demand function and the horizontal line at \( p = 100 \).

Step-by-step solution

Evaluate Demand for x=75

Substitute \( x = 75 \) into the equation to find the corresponding price: \( p = 3000 \left(1-\frac{5}{5+e^{-0.015 \cdot 75}}\right) \). Solve this expression for \( p \).

Evaluate Demand for x=200

Now Substitute \( x = 200 \) to find another pricing point: \( p = 3000 \left(1-\frac{5}{5+e^{-0.015 \cdot 200}}\right) \). Again, compute the result to find the price \( p \).

Graph the Demand Function

Plot the function \( p(x) = 3000 \left(1-\frac{5}{5+e^{-0.015x}}\right) \). Also, draw a horizontal line where \( p = 100 \) to visualize the price point for part (d).

Approximate Demand for Price \$ 100 From the Graph

Locate the point(s) of intersection between the graph of the demand function and the horizontal line at \( p = 100 \). Read off the corresponding \( x \) value(s) for demand.

Key concepts

Price Evaluation

Price evaluation is essential in understanding how much a consumer should expect to pay given a certain demand level. In the context of this exercise, evaluating the demand function allows us to determine the price at which a limited edition comic book can be sold when a specific number of units are demanded.

To evaluate the price, we substitute a given demand value into the demand function's equation. For instance, with a demand of 75 units, we substitute 75 into the demand equation, like so: \[ p = 3000 \left(1-\frac{5}{5+e^{-0.015 \cdot 75}}\right) \]This approach gives us the price for that specific demand.

This process is repeated for other demand values, such as 200 units, to get additional pricing information from the demand function.

Graphing Utility

Using a graphing utility is a practical way to visually analyze a function like the demand function for the comic book. A graphing utility, which can be a calculator, computer software, or an app, helps plot functions and provides a clear picture of how the price changes as demand varies.

When graphing the demand function:

• Plot the demand function equation: \[ p(x) = 3000 \left(1-\frac{5}{5+e^{-0.015x}}\right) \]
• Overlay a horizontal line on the graph at specific price points, such as \( p = 100 \).

This graphical representation aids in understanding the relationship between price and demand and can make it easier to approximate demand for given price levels.

Exponential Functions

Exponential functions are a class of mathematical functions characterized by their constant base raised to a variable exponent. These functions are versatile and often appear in various economic and scientific models.

In this exercise, the demand function contains an exponential term, \( e^{-0.015x} \), which is part of the denominator's equation within the price expression: \[ p(x) = 3000 \left(1-\frac{5}{5+e^{-0.015x}}\right) \]

The exponential term significantly influences the function's behavior, especially as the demand \( x \) changes. Small changes in \( x \) can lead to significant variations in the price, especially for lower values of demand.

Economic Applications

The demand function is a fundamental economic tool that helps businesses and economists understand consumer behavior and price dynamics. By analyzing these functions, businesses can make informed decisions about pricing strategies to maximize profits while meeting consumer demand.

This specific demand function for a limited edition comic book illustrates how price adjusts with varying demand levels, informing decisions such as:

• Setting appropriate price points for different demand scenarios.
• Forecasting how changes in market demand affect pricing.
• Identifying optimal production levels to match consumer interest.

Understanding these dynamics can guide businesses in aligning their strategies with market expectations and competitive pricing trends.