The number 
\[N(t)\] of trucks of a certain model that were sold within 
\[t\] months satisfies the following logistic differential equation:
 
 
\[\dfrac{dN}{dt}= N\cdot\left(90{,}000-\dfrac{3 N}{20{,}000} \right)\]
There were initially 
\[20{,}000{,}000\] trucks sold.
What is the carrying capacity of trucks sold?

The given logistic differential equation is:

\[
\frac{dN}{dt} = N \cdot \left(90{,}000 - \frac{3N}{20{,}000} \right)
\]

This is a logistic growth equation, and the general form of a logistic growth model is:

\[
\frac{dN}{dt} = r N \left(K - N \right)
\]

Where:
- \(r\) is the growth rate,
- \(K\) is the **carrying capacity**, the maximum population (in this case, trucks sold) the system can support,
- \(N\) is the population at time \(t\) (in this case, the number of trucks sold).

In the given equation:
- \(r = 90{,}000\), the growth rate,
- \(\frac{3}{20{,}000} N\) is the term limiting growth as \(N\) approaches the carrying capacity.

Now, let's rewrite the equation and focus on the part that determines the carrying capacity:

\[
\frac{dN}{dt} = N \cdot \left(90{,}000 - \frac{3}{20{,}000} N \right)
\]

At carrying capacity, the population stops growing, i.e., \(\frac{dN}{dt} = 0\). This occurs when the term inside the parentheses equals zero:

\[
90{,}000 - \frac{3}{20{,}000} N = 0
\]

Solving for \(N\):

\[
\frac{3}{20{,}000} N = 90{,}000
\]
\[
N = \frac{90{,}000 \times 20{,}000}{3}
\]
\[
N = 600{,}000{,}000
\]

Thus, the **carrying capacity** is \(600{,}000{,}000\) trucks.

Would you like further details or clarifications? Here are five related questions for deeper insight:

1. How does the logistic growth model apply to real-world population growth?
2. What is the significance of the growth rate \(r\) in logistic equations?
3. How can we solve the logistic differential equation for \(N(t)\)?
4. What happens to the growth rate as \(N\) approaches the carrying capacity?
5. How does the initial condition affect the behavior of logistic growth?

**Tip:** The carrying capacity in logistic growth is the point where growth stabilizes due to limiting factors like resources or market saturation.

The number N(t) of trucks of a certain model that were sold within t months satisfies the following logistic differential equation: 

dN/dt = N · (90,000 - (3N)/20,000). 
There were initially 20,000,000 trucks sold. What is the carrying capacity of trucks sold?

Explore how to determine the carrying capacity of truck sales using a logistic differential equation. This solution walks through key steps, including solving dN/dt = N(90,000 - 3N/20,000) and calculating the carrying capacity of 600,000,000 trucks. Suitable for students at an undergraduate level studying calculus and differential equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation