Please solve a) to c) correctly

Question

A bacteria culture starts with  bacteria and grows at a rate proportional to its size. After  hours there will be  bacteria.

(a) Express the population after  hours as a function of .

population:__________________________ (function of t)

(b) What will be the population after  hours?

(c) How long will it take for the population to reach  ?

Answer 1

Solution:

a) The general equation for population growth is the exponential function

y = P_i e ^kt , where P_i is the initial population, in your problem = 960

y is the final population also given as 3840 and time t = 4 hrs

substituting 3840 = 960 e^{k (4)} , then e^4k = 3840/960 = 4

Taking Ln of both sides Ln e^4k = ln 4, and 4k = ln4 , k= ln4/4 = 0.346

the equation for the exponential growth in terms of t is y = 960 e^{0.346 t}

b) After 6 hrs the population is y(6) = 960 e^{0.346 (6)} = 7653

c) To reach 2930 , then y = 2930 = 960 e^{0.346 t} , 3.052 = e^{0.346 t} Take ln of both sides

Ln 3.052 = Ln e^{0.346 t} , Ln 3.052 = 0.346 t , t = Ln 3.052 / 0.346 = 3.2 hrs