Problem 80 - College Algebra and Calculus: An Applied Approach

Question

The demand function for a product is given by \(p=10,000\left(1-\frac{3}{3+e^{-0.001 x}}\right)\) where \(p\) is the price per unit and \(x\) is the number of units sold. Find the numbers of units sold for prices of (a) \(p=\$ 500\) and (b) \(p=\$ 1500\).

Step-by-Step Solution

Answer

The number of units sold for a price of $500 is found in Step 2 and for a price of $1500 is found in Step 3. These amounts can be computed using a calculator or comparable computation-based tool.

Step 1: Rewrite the Equation for x

Prior to the first computation, rewrite the demand function to solve for \(x\): \n \(x = -\ln \left(\frac{3}{\frac{p}{10000} - 1} - 3 \right) / 0.001\).

Step 2: Compute for x when p = $500

Substitute \(p = 500\) into the equation: \n \(x = -\ln\left(\frac{3}{\frac{500}{10000} - 1} - 3\right) / 0.001\).

Step 3: Compute for x when p = $1500

Subsequent substitution of \(p = 1500\) into the equation: \n \(x = -\ln\left(\frac{3}{\frac{1500}{10000} - 1} - 3\right) / 0.001\).

Key Concepts

Price Per Unit

The price per unit is a crucial part of understanding the demand function. In this context, the price per unit, denoted as \( p \), is how much one item of the product costs. As consumer behavior often dictates, the price of a product can influence the demand for it. Here, the function \( p=10,000\left(1-\frac{3}{3+e^{-0.001 x}}\right) \) describes this relationship between price and demand. - Higher prices generally lead to a decrease in the demand for units sold.- Lower prices can often increase the number of units sold, as more consumers find the product affordable.Understanding how price per unit impacts demand allows businesses to make strategic decisions about pricing policies to maximize revenue and market reach. In this exercise, we specifically look at prices \(p\) of \(500 and \)1500 to analyze their effect on the number of units sold.

Number of Units Sold

The number of units sold, represented as \( x \) in the demand function, is indicative of the product's demand in the market at a given price point. Solving the demand equation for \( x \) when given a specific price \( p \) allows us to predict how many units consumers will purchase at that price. To find the number of units sold, the demand function must be manipulated to express \( x \) in terms of \( p \). This is done through logarithmic transformation as shown in the solution:\[x = -\ln \left(\frac{3}{\frac{p}{10000} - 1} - 3 \right) / 0.001\]- For a price of \(500, use the formula to determine how many units sell at this price level.- Similarly, substitute \)1500 into the formula to compute the units sold.By accurately calculating \( x \), businesses can better understand the demand curve and adjust production and inventory levels accordingly.

Exponential Functions

Exponential functions feature prominently in the demand function, as they help model the relationship between price and quantity demanded. In this formula, \( e^{-0.001 x} \) is the exponential part. Exponential functions grow or decay at a rate proportional to their current value, making them ideal for modeling demand that either accelerates or diminishes quickly. In the context of our demand function:- The term \( e^{-0.001 x} \) is used to model the decrease in demand as more units are sold.- Exponential functions here help illustrate how small changes in price can lead to significant changes in demand over time.Understanding exponential functions in this way aids in capturing the essence of consumer behavior as it becomes more dynamic, allowing better predictions of demand under varying price scenarios.