Question 1188652
.
A bacteria culture starts with 280 bacteria and grows at a rate proportional to its size. 
After 5 hours there will be 1400 bacteria.
A. What will the population be after 4 hours?
B. How long will it take the population to reach 2630? Give your answer accurate to 2 decimal places.
~~~~~~~~~~~~~~~~~~


<pre>
Write the exponential growth function with arbitrary base b

    B(t) = {{{280*b^t}}},

where "t" is the time in hours.


I use the multiplier 280, because it is the original number of bacteria at t = 0.


To find the base "b" value, write an equation as you read the problem

    1400 = {{{280*b^5}}},

or

    {{{1400/280}}} = {{{b^5}}}

        5  = {{{b^5}}}.


From here, you find  b = {{{root(5,5)}}} = 1.37973  (rounded).


So, the exponential growth function is  

    B(t) = {{{280*1.37973^t}}}.


To answer first question, substitute t= 4 into the formula  B(4) = {{{280*1.37973^4}}} = 1015 (rounded).


To answer the second question, solve an equation

    2630 = {{{280*1.37973^t}}}.

    {{{2630/280}}} = {{{1.37973^t}}}

    9.3929 = {{{1.37973^t}}}

    log(9.3929) = t*log(1.37973)

    t = {{{log((9.3929))/log((1.37973))}}} = 6.96 hours (about 7 hours).
</pre>

Solved.


-----------------


To see many other similar and different solved problems on bacteria growth, &nbsp;look into the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/logarithm/Bacteria-growth-problems.lesson>Bacteria growth problems</A> 

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lesson is the part of this online textbook under the topic "<U>Logarithms</U>".



Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.