Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Universal Instruments found that the monthly demand for its new line of Galaxy Home Computers \(t\) mo after placing the line on the market was given by $$ D(t)=2000-1500 e^{-0.05 t} \quad(t>0) $$ Graph this function and answer the following questions: a. What is the demand after 1 mo? After 1 yr? After 2 yr? After \(5 \mathrm{yr}\) ? b. At what level is the demand expected to stabilize?

Step-by-Step Solution

Answer

The demand after 1 month is \(2000 - 1500 e^{-0.05 (1)}\), after 1 year is \(2000 - 1500 e^{-0.05 (12)}\), after 2 years is \(2000 - 1500 e^{-0.05 (24)}\), and after 5 years is \(2000 - 1500 e^{-0.05 (60)}\). The demand is expected to stabilize at 2000.

Step 1: Understand the function

The given function for demand \(D(t)\) is \(D(t) = 2000 - 1500 e^{-0.05 t}\), where \(t\) represents the number of months after placing the line on the market.

Step 2: Calculate the demand at different time points

Given the demand function D(t), we can calculate the demand for different periods - 1 month, 1 year, 2 years, and 5 years. To do this, plug in the respective values of time (t) into the equation and compute the values: 1. For 1 month: \(t = 1\): \(D(1) = 2000 - 1500 e^{-0.05 (1)}\) 2. For 1 year: \(t = 12\) (1 year = 12 months): \(D(12) = 2000 - 1500 e^{-0.05 (12)}\) 3. For 2 years: \(t = 24\) (2 years = 24 months): \(D(24) = 2000 - 1500 e^{-0.05 (24)}\) 4. For 5 years: \(t = 60\) (5 years = 60 months): \(D(60) = 2000 - 1500 e^{-0.05 (60)}\)

Step 3: Graph the function

Using a graphing tool (calculator, computer software, or online tools), graph the function \(D(t) = 2000 - 1500 e^{-0.05t}\) for \(t>0\). Pay attention to the shape of the graph and how it behaves as t increases.

Step 4: Finding the level at which demand stabilizes

From the shape of the graph, we can see that as \(t\to \infty\), the demand function \(D(t)\) approaches a horizontal asymptote, which represents the stable level of demand. To find this stable level, we need to find the limit of the function as \(t\to \infty\): \(\lim_{t\to\infty} D(t) = \lim_{t\to \infty} (2000 - 1500 e^{-0.05 t})\) As \(t\to \infty\), the exponent part \(e^{-0.05t}\) goes to 0. Thus, the limit becomes: \(\lim_{t\to\infty} D(t) = 2000\) So the demand is expected to stabilize at 2000.

Key Concepts

Demand Forecasting

Demand forecasting is an essential activity businesses engage in to predict future consumer demand for their products. This allows companies to efficiently manage their resources and supply chain. In the context of Universal Instruments’ Galaxy Home Computers, the demand function helps forecast how many units will be sold over time.
The function given is an example of exponential smoothing:

• It considers initial demand but also factors the decrease in growth rate over time with the term \(-1500 e^{-0.05 t}\).
• The parameter \(e^{-0.05 t}\) indicates how rapidly demand is expected to change with time.
  
• It also shows that after the initial market excitement declines, the product stabilizes at a predictable level.
  

For effective forecasting, businesses analyze historical data and external conditions that might affect sales, such as market trends, economic factors, and consumer preferences.

Asymptotes

In the realm of functions, an asymptote is a line that a graph approaches but never actually reaches. For the given demand function \(D(t) = 2000 - 1500 e^{-0.05 t}\), understanding its asymptotic behavior is crucial for predicting long-term outcomes.
The term \(2000\) in the function represents a horizontal asymptote.

• As time \(t\to \infty\), the exponential component \(e^{-0.05 t}\) tends towards zero, making the demand approach 2000 units.

This indicates that while the demand will fluctuate initially, it will stabilize at a level determined by the asymptote.
Asymptotic behavior is a powerful tool for businesses, as it suggests what the stable, long-term demand will be, guiding decisions on production and marketing.

Graphing Functions

Graphing functions is an integral part of understanding their behavior and form. For the function \(D(t) = 2000 - 1500 e^{-0.05 t}\), graphing aids in visually interpreting demand changes over time.
To graph this:

• Start by identifying key points, such as the initial point \(t=0\), where \(D(0)=500\).
• As time progresses, plot points like \(t=1\), \(t=12\), and other calculated values.
• Examine how the exponential decay term \(e^{-0.05 t}\) affects the curve, causing it to gradually approach the horizontal asymptote at \(D(t) = 2000\).
  

Graphing provides a clear picture, revealing how demand initially increases quickly and then stabilizes.
By visualizing the graph, students and businesses can better comprehend how different parameters in the function influence demand over time. This process can further enhance intuitive understanding and practical application.