Question

Compact florescent light bulbs save energy when compared to traditional incandescent bulbs. Our green energy campaign includes efforts to get local residents to change their incandescent bulbs to fluorescent bulbs. Initially 200 households make the change. Market studies suggest that, in the absence of limiting factors, we could increase that number by 25% each month. In our target area, there are 252,200 households, which we take as the limiting value.

Required:
Make a logistic model that gives the number of households converting to florescent bulbs after t months. (Use t as your variable. Round r to three decimal places.)

Answer 1

The logistic equation that models this situation is given by:

$P(t) = \frac{252200}{1 + 1260e^{-0.25t}}$

The logistic equation for a population is given by:

$P(t) = \frac{K}{1 + Ae^{-kt}}$

$A = \frac{K - P(0)}{P(0)}$

In which:

• K is the carrying capacity.
• P(0) is the initial value.
• k is the growth rate, as a decimal.

In this problem:

• Total of 252,200 households, which means that this is the maximum number of households that may make the change, hence, $K = 252200$.
• Increase of 25% each month, hence, $k = 0.25$.
• Initially, 200 households made the change, hence $P(0) = 200$.

Then:

$A = \frac{252200 - 200}{200} = 1260$

$P(t) = \frac{252200}{1 + 1260e^{-0.25t}}$

A similar problem is given at https://brainly.com/question/13229117