Problem 65 - College Algebra and Calculus: An Applied Approach

Question

Demand A manufacturing company forecasts that the demand \(x\) (in units per year) for its product over the next 10 years can be modeled by \(x=500\left(20+t e^{-0.1 t}\right)\) for \(0 \leq t \leq 10\), where \(t\) is the time in years. (T) (a) Use a graphing utility to decide whether the company is forecasting an increase or a decrease in demand over the decade. (b) According to the model, what is the total demand over the next 10 years? (c) Find the average annual demand during the 10 -year period.

Step-by-Step Solution

Answer

Based on the graph, the company is forecasting an increase in demand over the next decade. The exact total and average annual demands would be numerical values based on the calculations from the integral and the simple division, respectively.

Step 1: Sketching the Graph of the Demand Function

Using a graphing utility, plot the function \(x = 500(20 + te^{-0.1t})\) for \(0 \leq t \leq 10\). Observe the behavior of the graph to determine whether the demand is increasing or decreasing.

Step 2: Calculating the Total Demand

To find the total demand over the next 10 years, integrate the demand function \(x = 500(20 + te^{-0.1t})\) over the interval [0, 10]. The total demand over the next 10 years is \(\int_{0}^{10} 500(20 + te^{-0.1t}) dt\).

Step 3: Calculating the Average Demand

Divide the total demand by the time period to find the average annual demand. The average demand during this 10-year period is the total demand divided by 10 given in the previous step.

Key Concepts

Graphing Utility

When exploring complex functions like demand equations, a graphing utility becomes an invaluable tool. In our example, the demand function is represented by the equation x = 500(20 + te^{-0.1t}), where t symbolizes time in years, and x represents the units demanded per year. By plotting this function on a graphing utility, students can visualize how the demand changes over a span of 10 years.

Using the graphing utility, you can observe the curve and its slope to determine whether the demand is predicted to increase or decrease over time. An upward slope suggests an increase in demand, while a downward slope indicates a decrease. Graphical representation makes it easier to comprehend the behavior of the function at a glance, which could be more challenging if tried to infer just from the formula itself.

Modeling Demand

Modeling demand is about representing consumer behavior mathematically. In our problem, the demand is modeled by a function that considers both a constant factor and a variable factor that changes over time. The equation x = 500(20 + te^{-0.1t}) includes a growth factor dictated by 20+t, and a decay factor expressed by e^{-0.1t}. This reflects a realistic approach to demand forecasting, as initial demand generally grows but may diminish over time due to various factors such as market saturation or product obsolescence.

By accurately modeling demand, businesses can make informed production and financial strategies, adjusting supply to meet the anticipated market needs. Therefore, having a good demand function can be as vital as the product itself in a company's success.

Integration Calculus

Integration in calculus is a mathematical method used to calculate the total amount, area, or volume over an interval. In terms of economics and modeling demand as we have in our example, integration allows us to find the total product demand over a period. By integrating the demand function x = 500(20 + te^{-0.1t}) from t = 0 to t = 10, we effectively sum up all the infinitesimal quantities of demand over the 10 years.

The integration calculus handles continuously changing quantities - precisely what happens with product demand over time. The integral, in this context, is not only a mathematical solution but also an essential concept for understanding total output, cost, and revenue in an economic setting.

Average Annual Demand

The average annual demand is a critical metric in understanding a product's market performance over a given period. It serves as a standardized way to compare the yearly demand without the fluctuations of shorter intervals. Once students have calculated the total demand through integration, as shown in Step 2, finding the average annual demand is straightforward.

To calculate, simply divide the total demand, obtained through integration of the demand function over 10 years, by the time period, which in this case is 10. This calculation gives the average amount of product demanded per year and normalizes the forecast, providing clarity to the company for production planning, inventory management, and overall strategy.