Respuesta :
Answer: It will take 11.56 hours .
Step-by-step explanation:
Exponential growth in population or size formula :
[tex]P(t)=P_0e^{rt}[/tex]
, where [tex]P_0[/tex] = initial size
r= rate of growth
t= time period
As per given , we have
[tex]P_0=10[/tex] grams
At t= 1 , P(t)= 11 grams
Then,
[tex]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[/tex]
When, the bacteria have tripled in size , P(t) = 3 x10 = 30
Then,
[tex]30=10e^{0.095t}\\\\ 3=e^{0.095t}[/tex]
[tex]\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[/tex]
Hence, it will take 11.56 hours .
A bacteria culture is initially 10 grams at t=0 hours & grows at a rate proportional to its size , After an hour the bacteria culture weighs 11 grams , The bacteria takes 11.56 hours to have tripled in size.
To find the time of bacteria when increasing the growth to tripled.
Given : when time=0 hours , weight=10 grams.
when time=1 hours , weight=11 grams.
To find: when time= ? hours , weight=30grams.
Here according to question, initial size = 10 grams we have asked for tripled in size i.e. 30 grams.
Now we knows that,
The formula for exponential growth in population or size is
[tex]\rm (P)=P_0e^{rt}[/tex] where,
[tex]\rm P_0=initial\;size\\\\r= rate\;of\;growth\\\\t= time \;period[/tex]
Now, we put the value in formula we get,
[tex]\rm P_0=10\;grams \\\\when ,\\\;\;t=1\;hour P(t)=11 grams\\Then,\\11=10e^{r(1)\\1.1 =e^r\\\\\rm Taking \;log(natural)\;both\;the\; side \;on \;solving\;we\;get,\\ln(1.1)=r\;ln(e)\\r=ln(1.1)\\r=0.953101798043\approx0.095[/tex]
Now when the bacteria increase its size to triple
[tex]\rm P(t) = 3 \times 10 = 30[/tex]
Then, according to the formula we substitute values in the formula,
[tex]\rm 30=10e^{0.095t}\\\\3=e^{0.095t}\\\\Again \;we \;take\;natural\;log\;on \;both\;the\;sides, we\;get\\ln\;3=0.095t\\\\t=\dfrac{\rm ln\;3}{0.095}\\\\\\\\\rm t= \dfrac{1.09861228867}{0.095} \\\\\ t=approx \; 11.56[/tex]
Therefore, The bacteria takes 11.56 hours to have tripled in size.
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