Calculus rate of change

Question

Suppose that the population of bacteria originally (at time t=0) numbers 2 million, and triples every hour, so the population is P(t)=2x3^t (million) bacteria after t hours. 

 

a.) Find how long it takes this bacteria population to quadruple (to 8 million)

 

b.) what is the rate of growth of this population after 4 hours

Answer 1

a. P(t) = 2(3)^t

 

    P(t) = 8 when 2(3)^t = 8

 

                            3^t = 4    ln(3^t) = ln4

 

                                         t(ln3) = ln4

 

                                               t = ln4 / ln3 ≈ 1.26 hours

 

b.  P'(t) = 2 (ln3)(3^t)

 

     Rate of growth after 4 hours = 2(ln3)(3⁴) = 178 million bacteria per hour