Problem 31 - Precalculus

Question

A new model BMW convertible coupe is designed and produced in time to appear in North America in the fall. BMW Corporation has a limited number of new models available. The number of new model BMW convertible coupes purchased in North America is given by \(N=\frac{100,000}{1+10 e^{-2 t}},\) where \(t\) is the number of weeks after the BMW is released. a. How many new model BMW convertible coupes will have been purchased 2 weeks after the new model becomes available? b. How many after 30 weeks? c. What is the maximum number of new model BMW convertible coupes that will be sold in North America?

Step-by-Step Solution

Answer

Approximately 8,554 cars will be purchased after 2 weeks, 100,000 after 30 weeks, and the maximum number sold is 100,000.

Step 1: Understanding the Function

The function provided is \( N = \frac{100,000}{1 + 10 e^{-2t}} \). This represents a logistic-like model often used to describe growth processes such as sales or population. In this context, \( N(t) \) is the cumulative number of cars purchased \( t \) weeks after the model is released.

Step 2: Solving for Week 2

To find out how many cars will have been purchased after 2 weeks, plug in \( t = 2 \) into the function. Calculate \( N(2) = \frac{100,000}{1 + 10 e^{-4}} \).

Step 3: Calculating Week 2 Expression

Calculate \( e^{-4} \) using a calculator, which is approximately 0.0183. Substitute this back into the expression: \( N(2) \approx \frac{100,000}{1 + 0.183} \).

Step 4: Result for Week 2

Simplify the calculation to get \( N(2) \approx \frac{100,000}{1.183} \approx 84,554 \). Therefore, approximately 8,554 cars will have been purchased after 2 weeks.

Step 5: Solving for Week 30

For 30 weeks, substitute \( t = 30 \) into the equation: \( N(30) = \frac{100,000}{1 + 10 e^{-60}} \).

Step 6: Calculating Week 30 Expression

As \( e^{-60} \) is extremely small, practically it approaches 0. Thus, the denominator becomes approximately 1, making \( N(30) \approx \frac{100,000}{1} = 100,000 \).

Step 7: Result for Week 30

After 30 weeks, about 100,000 cars will have been purchased.

Step 8: Determining the Maximum Sales

As \( t \to \infty \), \( e^{-2t} \to 0 \). Thus, the denominator approaches 1, making the function reach \( N = 100,000 \). This represents the maximum number of cars that can be sold.

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. This type of function is widespread in modeling growth and decay processes. In our example, the exponential component is represented by \( e^{-2t} \). Here, the base \( e \) is Euler's number, an important mathematical constant approximately equal to 2.71828. The exponent \(-2t\) illustrates how the rate of change affects sales over time.

Exponential functions have a few key properties:

• Growth or decay rate: Positive exponents result in growth, while negative exponents, like in our model, indicate decay.
• Rapid initial change: The amount of change is significant in the early periods (smaller \( t \) values), tapering off over time.
• Asymptotic behavior: As \( t \) approaches large values, \( e^{-2t} \) approaches zero, causing the denominator in our function \( 1 + 10e^{-2t} \) to stabilize at 1.

These features make the exponential function essential for describing dynamic systems like population or sales growth, where the rates of change vary significantly over time.

Cumulative Sales

Cumulative sales refer to the total number of units sold by a certain time period. In our scenario, \( N(t) \) represents the cumulative sales of the BMW model 2 weeks and 30 weeks after its release. This is crucial for understanding business performance over time.

The function \( N = \frac{100,000}{1+10e^{-2t}} \) helps trace sales over weeks, showing how purchases accumulate:

• Short-term analysis: For example, at 2 weeks \( t=2 \), the calculation demonstrated around 8,554 cars sold.
• Long-term observation: At 30 weeks \( t=30 \), sales reached near saturation with 100,000 cars sold, demonstrating the effectiveness of the model's marketing strategy.
• Sales progression: This model shows how sales accelerate early and then level off as time passes and sales capacity is nearly reached.

Understanding cumulative sales through such models allows businesses to anticipate demand better, plan production, and manage inventory efficiently.

Population Growth Model

The BMW sales model is akin to a population growth model, exemplified by logistic growth. This kind of growth is especially relevant when resources or environments have limitations.

Key features of logistic growth models include:

• Carrying capacity: Similar to how a maximum population exists due to limited resources, the BMW model caps at 100,000 cars, the maximum number expected to be sold.
• Rapid initial growth: At the start, sales (or population) rise sharply as the market absorbs most available units quickly.
• Slowing growth: As the market approaches saturation (near carrying capacity), the growth rate naturally decreases since fewer potential customers remain or (analogous to populations) resources become stretched.

This approach shows the practicality and realism of using a logistic growth model. It reflects the real-world dynamics where restrictions invariably apply, providing an insightful method for planning and policy-making in varied contexts like sales, population management, or even technology adoption.