When will the population reach 20,000?

Question

A bacteria culture initially contains 500 cells and grows a rate proportional to its size. After an hour the population has increased to 1200.

a). Find the expression for the number of bacteria after t hours. Answer: k= 0.014591146 because

1200= 500$e^(k*60)$

divide by 1200/500 and then do ln|| on both sides to get k.

b) Find the number of bacteria after 4 hours? P(240)= 500$e^(0.014591146*240)$

Answer: 16,588

c) Find the rate of growth.

dy/dt= 0.014591146*16588= 242/60 = 4.03 cells/hr

d) When will the population reach 20,000?

The way I went about this is: $$20,000 = 500e^{0.014591146\,t}$$

which turned into $\ln|40|= \ln|e^{0.014591146\,t}|$

*The $\ln|e^x|$ cancels so I am left with

$$\frac{\ln|40|}{0.014591146} =t$$

My answer came out to be 252 years? However my professor said my answer was incorrect? What am I doing wrong?

Answer 1

The problem says that initially there are 500 cells and after an hour the population has increased to 1200. To find $k= 0.014591146$ you plug $t=60$ so $k$ is measured in 1/minute. The precise result t=252.81629380680148789220913131845 is the number of MINUTES not YEARS.

Answer 2

Using your Maths, we get:

$$t=252.81629380680148789220913131845$$

and this should be the answer.

If we round to years, we need the population to exceed $20,000$, and $252$ falls short, only yielding $19764$, and so we need $253$ years.