Question

MAKE A DECISION: DEMAND FUNCTION A company that produces cell phones estimates that the demand \(D\) for a new model of phone is given by \(D=-x^{3}+54 x^{2}-140 x-3000, \quad 10 \leq x \leq 50\) where \(x\) is the price of the phone (in dollars). (a) Use a graphing utility to graph \(D\). Use the trace feature to determine the values of \(x\) for which the demand is 14,400 phones. (b) You may also determine the values of \(x\) for which the demand is 14,400 phones by setting \(D\) equal to 14,400 and solving for \(x\) with a graphing utility. Discuss this alternative solution method. Of the solutions that lie within the given interval, what price would you recommend the company charge for the phones?

Short answer

Using the trace feature on the graph of the function, the two prices that correspond to a demand of 14,400 units can be found. The recommendation will depends on external market factors, but typically a company would want to set the higher price to generate more revenue.

Step-by-step solution

Understand The Problem

Analyze the given problem. The demand \(D\) is a function of \(x\) (the price), which is a cubic polynomial. The goal is to find out the price(s) for which the demand is 14,400 units. Given it's much too complex to solve algebraically, the problem suggests to solve it graphically using a graphing utility.

Graph The Function

First, graph the function \(D = -x^{3}+54x^{2}-140x-3000\) over the interval [10,50]. This can be done using a graphing calculator or a software program that can graph equation.

Trace the Function for Demand = 14,400

Now, input 14,400 into the trace feature of the graphing utility. This feature allows you to input a 'y' value and it will return the corresponding 'x' values (the prices in our case).

Identify the Prices

According to the graph, two prices correspond to a demand of 14,400 units. These prices need to be noted down. Remember to only note down the values within the given interval [10,50]

Relate to Real-world Situation

Use common business sense to determine which of these two prices the company should charge for the phones. Usually, businesses would choose the higher price because this generates higher revenue. However, factors such as competition and market conditions might lead the company to choose the lower price.

Key concepts

Cubic Polynomial

When discussing demand functions, it often involves the use of polynomial equations. In this specific exercise, we are examining a cubic polynomial. A cubic polynomial is defined as a polynomial of degree three, meaning it has the general form of \(ax^3 + bx^2 + cx + d\), where \(a\), \(b\), \(c\), and \(d\) are constants, and \(a eq 0\).
In our example, the demand \(D\) for the phones is expressed as a cubic polynomial with \(D = -x^3 + 54x^2 - 140x - 3000\).
This equation helps in modeling the relationship between the price \(x\) and the demand \(D\). The coefficients of \(x\) determine the shape and behavior of the graph, impacting how prices affect demand.

Graphing Utility

A graphing utility is a tool, often in the form of a calculator or software, that allows us to visualize mathematical equations. It's especially useful for solving complex equations graphically rather than algebraically.
In this problem, we use a graphing utility to plot the demand function \(D = -x^3 + 54x^2 - 140x - 3000\) and find the price corresponding to a specific demand level.
By seeing the graph, we can understand the behavior of the polynomial across the given interval, which is from \(x = 10\) to \(x = 50\). Using the trace feature of the graphing utility enables us to input a specific demand (14,400 units) and retrieve the price values that meet this demand.

Price-Demand Analysis

In economics, price-demand analysis is crucial for setting effective pricing strategies. It involves understanding how different price levels affect consumer demand for a product.
In the given exercise, the goal is to determine which price point within the interval [10,50] achieves a demand of 14,400 phones. There are often multiple price points that result in a specific demand due to the complex nature of polynomial demand functions.
Once the prices are identified through the graph, a company must analyze which price makes the most sense in a real-world scenario. Considerations include maximizing revenue without scaring off customers due to high prices, or underpricing and leaving money on the table. External factors like market competition, cost of production, and consumer income also play a role.

Graphical Solution Method

Graphical solutions offer a way to solve equations that might be too complex for algebraic methods. By visualizing the function graphically, we can immediately see where the function meets specific criteria.
For our demand function, plotting the graph allows us to see at what price levels the demand equals 14,400 units. This method is advantageous as it provides a visual understanding of the interplay between price and demand and identifies solutions quickly.
To carry out a graphical solution, we set the demand \(D\) to 14,400 and find corresponding \(x\)-values on the graph. In scenarios where precision is necessary, such as choosing between two prices, additional consideration for practicality beyond the graph may be involved, like market conditions and business strategy.