Integrate N'(t): N(t)=c+(50/k)e^(kt). N'(5)=50e^(5k)=150, so e^5k=3, 5k=ln(3), k=ln(3)/5=0.2197.
Initially at t=0 there are 100 cells, so N(0)=c+250/ln(3)=100; c=100-250/ln(3)=-127.56. Therefore:
N(t)=250(e^(ln(3)t/5)-1)/ln(3)+100.
When t=12, N(12)=250(e^(2.4ln(3))-1)+100
ln(3)=1.0986 approx, so N(12)=250(e^(2.4*1.0986)-1)/1.0986+100=3051 cells.
[When t=5, N(5)=555 cells.]