Question

A bacteria culture is initially 10 grams at t=0 hours and grows at a rate proportional to its size. After an hour the bacteria culture weighs 11 grams. At what time will the bacteria have tripled in size?

Answer 1

Answer: It will take 11.56 hours .

Step-by-step explanation:

Exponential growth in population or size formula :

$P(t)=P_0e^{rt}$

, where $P_0$ = initial size

r= rate of growth

t= time period

As per given , we have

$P_0=10$ grams

At t= 1 , P(t)= 11 grams

Then,

$11=10e^{r(1)}\\\\ 1.1= e^r\\\\\text{Taking natural log on both sides , we get} \\\\\ln (1.1)=r\ln (e)\\\\ r=\ln (1.1)\\\\ r=0.0953101798043\approx0.095$

When, the  bacteria have tripled in size , P(t) = 3 x10 = 30

Then,

$30=10e^{0.095t}\\\\ 3=e^{0.095t}$

$\text{Taking natural log on both sides , we get}\\\\ \ln 3=0.095t\\\\ t=\dfrac{\ln3}{0.095}\\\\ t=\dfrac{1.09861228867}{0.095}\approx11.56$

Hence, it will take 11.56 hours .