Problem 58 - College Algebra and Calculus: An Applied Approach

Question

The demand function for a home theater sound system is given by \(p=7500\left(1-\frac{7}{7+e^{-0.003 x}}\right)\) (a) Find the price \(p\) for a demand of \(x=200\) units. (b) Find the price \(p\) for a demand of \(x=900\) 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 \(\$ 400\).

Step-by-Step Solution

Answer

The price for a demand of 200 units is calculated through the demand function. Similarly, the price for 900 units is calculated. A graphing utility produces the graph of the function. Using the graph, we estimate the demand when the price is $400.

Step 1: Substitute X into the Demand Function

To find the price for a specific demand, we substitute the given value of \(x\) into the demand function. For \(x=200\) units, the price is calculated as: \(p=7500\left(1-\frac{7}{7+e^{-0.003*200}}\right)\)

Step 2: Perform the calculation

By performing the calculation in the previous step, we get the price of the system when \(x=200\) units.

Step 3: Repeat for different demand

We now repeat the process for \(x=900\) units, substituting \(x\) into the demand function and performing the calculation again:\(p=7500\left(1-\frac{7}{7+e^{-0.003*900}}\right)\)

Step 4: Use a graphing utility to graph the function

Using a graphing utility, we graph the function \(p=7500\left(1-\frac{7}{7+e^{-0.003*x}}\right)\). Make sure to label the x-axis as 'Demand' and the y-axis as 'Price'.

Step 5: Use the graph to approximate demand

By using the graph produced in Step 4, we find the demand when the price is $400.

Key Concepts

Price Elasticity of Demand

Understanding the price elasticity of demand is critical for businesses and economists as it measures how sensitive the quantity demanded of a product is to a change in its price. It's a ratio that shows the percentage change in quantity demanded in response to a one percent change in price.

To calculate the price elasticity of demand (PED), we use the formula: \[PED = \dfrac{\% \text{ Change in Quantity Demanded}}{\% \text{ Change in Price}}\]In the context of the provided exercise, where the demand function is an exponential function, one could calculate the PED at different points to see how demand is affected by price changes at different levels of demand.

Generally, if PED is greater than 1, demand is elastic, meaning consumers are highly responsive to price changes. If it is less than 1, demand is inelastic. For instance, if the PED for the home theater sound system is found to be greater than 1 at a certain price level, a small decrease in price could lead to a significant increase in the amount of units demanded, which may be a strategy for a company to increase sales volume.

Graphing Utility in Algebra

A graphing utility is an invaluable tool in algebra for visualizing complex functions and understanding the relationship between variables. It can be as simple as a graphing calculator or as sophisticated as computer software.

For example, to graph the demand function from the exercise, the graphing utility plots the price (\(p\)) on the y-axis against the demand (\(x\)) on the x-axis. By plotting various values of \(x\), the curve of the demand function is formed, illustrating how price varies with demand.

One can use the graph to approximate values of \(x\) for a given \(p\) (such as determining the demand when price is \(\$400\)), or conversely, to estimate \(p\) for a certain \(x\). This visual representation helps in understanding the interaction between variables, predicting outcomes, and making informed decisions. Graphs can also reveal properties such as intercepts, asymptotes, and areas of elasticity or inelasticity.

Exponential Functions

An exponential function is a mathematical expression where a variable appears as an exponent. In real-world scenarios, they often model growth or decay processes such as population growth, radioactive decay, or, as in our example, changes in demand relative to price.

The general form of an exponential function is:\[ f(x) = ab^{x}\]where \(a\) is a constant, \(b\) is the base of the exponential function, and \(x\) is the exponent. In the exercise, the demand function is given by:\[p=7500\left(1-\frac{7}{7+e^{-0.003 x}}\right)\]This function represents an exponential decay model, where as \(x\) (the quantity demanded) increases, the term \(e^{-0.003x}\) decreases, hence the price \(p\) approaches a certain value — in this case, \(7500\).

Understanding exponential functions allows us to analyze how small changes in the exponent can significantly affect the outcome, which is pivotal for economic modeling and interpretations of supply and demand.