Answer:
in order to triple the inicial population of spiders, will take 50395 days
Step-by-step explanation:
we can define the termite population function as T(t) and the one for spiders as S(t) , where t represents time measured in days
since both have and exponencial growth
T(t)= a*e^(b*t)
S(t)= c*e^(d*t)
1) when the day the person moves in , t=0 and T(0)= 120 termites
T(0) = a*e^(b*0) = a = 120
2) after 4 days , t=4 and the house contains T(4) = 210 termites
T(4)= 120*e^(b*4) = 210 → 4*b = ln (210/120) → b = (1/4)* ln(210/120)= 0.14
therefore
T(t) = 120*e^(0.14*t)
3) 3 days after moving in , t=3, there were T(3) = 120*e^(0.14*3)=182.63≈ 182 termites . The number of spiders is half of the number of termites → S(3) = T(3) * 1 spider/ 2 termites =91.31 spiders ≈ 91 spiders
4) after 8 days of moving in , t=8, there were T(8) = 120*e^(0.14*8)=367.78≈ 368 termites . The number of spiders is 0.25 times the number of termites → S(8) = T(8) * 1 spider/ 4 termites =91.94 spiders ≈ 92 spiders
from
S(t)= c*e^(d*t) → d = ln [S (tb)/S (ta) ] / (tb-ta)
therefore d = ln [ S(8)/S(3) ] / (8 - 3 ) = 2.18*10^-5
in order to triple the initial population
S(t3) = 3 *S(0) = 3*[c*e^(d*0)] = 3*c
S(t3) = c*e^(d*t3) = 3* c → t3 = ln(3) / d = 50395 days
