**Magazine Sales Our new magazine initially sells 300 copies per month. Research indicates that a vigorous advertising campaign could increase sales by 20% each month if our market were unlimited. But research also indicates that magazine sales in our area are unlikely to exceed 1200 per month. Make a logistic model of projected magazine sales.**

To model the magazine sales, we can use a **logistic growth model**. This model is suitable here because it allows for an initial growth rate that eventually levels off as the sales approach a maximum, or "carrying capacity," due to market limitations.

### Setting Up the Logistic Model
The logistic growth model is typically given by:

\[
S(t) = \frac{K}{1 + \frac{K - S_0}{S_0} e^{-rt}}
\]

where:
- \( S(t) \) is the sales projection at time \( t \) (in months),
- \( K \) is the carrying capacity (maximum possible sales),
- \( S_0 \) is the initial sales,
- \( r \) is the growth rate,
- \( t \) is the time in months.

### Given Information
- **Initial sales \( S_0 = 300 \) copies per month**,
- **Carrying capacity \( K = 1200 \) copies per month**,
- **Monthly growth rate without limitation**: \(20\%\), which gives us \( r = 0.20 \).

### Building the Model
Substituting the known values:

\[
S(t) = \frac{1200}{1 + \frac{1200 - 300}{300} e^{-0.20t}}
\]

#### Simplification of the Term Inside the Denominator
1. Calculate the ratio \( \frac{1200 - 300}{300} \):
   \[
   \frac{1200 - 300}{300} = \frac{900}{300} = 3
   \]
   
2. Substituting back, the model becomes:
   \[
   S(t) = \frac{1200}{1 + 3 e^{-0.20t}}
   \]

### Final Logistic Model
The projected magazine sales after \( t \) months is given by:
\[
S(t) = \frac{1200}{1 + 3 e^{-0.20t}}
\]

This function models the growth of magazine sales, accounting for both the initial increase in sales due to advertising and the market saturation at 1200 copies.

Would you like a breakdown of the calculation steps or have any other questions?

---

1. How does the logistic model behave as \( t \to \infty \)?
2. What would the model predict for sales after 6 months?
3. How does changing the growth rate \( r \) impact the logistic model?
4. If the carrying capacity increased to 1500, how would the model change?
5. What if the initial sales were only 200 copies?

**Tip:** Logistic models are often used in scenarios with growth limitations due to resource constraints or market saturation.

Magazine Sales Our new magazine initially sells 300 copies per month. Research indicates that a vigorous advertising campaign could increase sales by 20% each month if our market were unlimited. But research also indicates that magazine sales in our area are unlikely to exceed 1200 per month. Make a logistic model of projected magazine sales.

Explore how to create a logistic growth model to project magazine sales, starting at 300 copies per month with a 20% growth rate, capped at a maximum of 1200 copies due to market saturation. Learn the step-by-step setup of the logistic equation S(t) = 1200 / (1 + 3 * e^(-0.20t)). Ideal for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation