The population p(t) at time t of a certain mouse species satisfies the differential equation d p(t) d t =0.5 p ( t )-450 . If p(0)=850 , then the time at which the population becomes zero is :

Step-by-step solution

Given differential equation is d p(t) d t =0.5 p(t)-450 d p(t) d t = 1 2 p(t)-450 d p(t) d t = p(t)-900 2 2 d p(t) d t =-[900-p(t)] 2 d p(t) 900-p(t) =-d t Integrate both the side, we get : -2 d p(t) 900-p(t) = d t Let 900-p(t)=u -d p(t)=d u We have; 2 d u u = d t 2 u=t+c 2 [900-p(t)]=t+c when t=0, p(0)=8502 (50)=c 2 [ ( 900-p(t) 50 ) ]=t 900-p(t)=50 e^ t 2 p(t)=900-50 e^ t 2 let p (t₁ )=00=900-50 e^ t₁ 2 t₁=2 18