The logistic growth model PT081167e016t represents the propo

The logistic growth model P(T)=0.8/1+1.67e^-0.16t represents the proportion of new cars with GPS.Let t=0 represent 2006,t=1represent2007, and so on.

A. what proportion of new cars in 2006 had GPS? answer_%

B. During what year will 75% of the new cars have GPS?

Solution

given P(t)=0.8_{/(1+1.67e^-0.16t)}

A)

t=0 represent 2006

P(0)=0.8_{/(1+1.67e^-0.16*0)}

P(0)=0.8_/(1+1.67)

P(0)=0.8_/2.67

P(0)=0.299625468164794

P(0)=29.96%

on rounding

P(0)=30%

30% of new cars in 2006 had GPS

B)

75% of the new cars have GPS

=>P(t)=75%

=>P(t)=0.75

=>0.8_{/(1+1.67e^-0.16*t)}=0.75

=>(1+1.67e^-0.16*t)=0.8_/0.75

=>1.67e^-0.16*t=(0.8_/0.75)-1

=>e^-0.16*t=(1_/1.67)((0.8_/0.75)-1)

=>-0.16*t=ln[(1_/1.67)((0.8_/0.75)-1)]

=>t=(-1_/0.16)ln[(1_/1.67)((0.8_/0.75)-1)]

=>t=20.1305

year =2006+20.1305

year =2026.1305

in year 2027 ,75% of the new cars will have GPS