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 florescent bulbs. Initially 184 households make the change. Market studies suggest that, in the absence of limiting factors, we could increase that number by 27% each month. In our target area, there are 232,024 households, which we take as the limiting value. 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.)

N(t) =

Answer 1

Initially 184 households make the change from the traditional incandescent bulbs to florescent light bulbs. Thereafter, this number increases by 27% each month. Thus, the numbers of the households adopting the change in the succeeding months are 184*1.27, 184*(1.27)² , 184*(1.27)³, 184*(1.27)⁴,……

Thus, the number of households converting to florescent bulbs after t months is N(t) = 184*(1.27)^t. Further, since the limiting value is 232024, hence if the entire target area of 232,024 households adopts the change, then 184*(1.27)^t = 232024 or, (1.27)^t =232024/184 = 1261.

Now, on taking logarithm of both the sides, we get t log 1.27 = log 1261 or, 0.103803721 t = 3.100715087 so that t = 3.100715087/0.103803721 = 29.871 ( on rounding off to three decimal places).