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?

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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.