SOLUTION: 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 hour

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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.
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
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.
~~~~~~~~~~~~~~~~~~ 

Write the exponential growth function with arbitrary base b

    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 = ,

or

     = 

        5  = .


From here, you find  b =  = 1.37973  (rounded).


So, the exponential growth function is  

    B(t) = .


To answer first question, substitute t= 4 into the formula  B(4) =  = 1015 (rounded).


To answer the second question, solve an equation

    2630 = .

     = 

    9.3929 = 

    log(9.3929) = t*log(1.37973)

    t =  = 6.96 hours (about 7 hours).

Solved.

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

To see many other similar and different solved problems on bacteria growth,  look into the lesson
    - Bacteria growth problems
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic "Logarithms".


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Free of charge online textbook in ALGEBRA-I
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to your archive and use it when it is needed.



Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
i think this means that it grows at a constant rate.
for example, something like 10% of its size, no matter what its size is.

the formula for that is f = p * (1 + r) ^ n

f is the future value.
p is the present value
r is the rate of growth per time period.
n is the number of time periods.

they state that it grows from 280 to 1400 in 5 hours.

the formula becomes 1400 = 280 * (1 + r) ^ 5.
divide both sides of this formula by 280 to get:
1400/280 = (1 + r) ^ 5.
take the fifth root of both sides of this equation to get:
(1400/280)^(1/5) = 1 + r.
subtract 1 from both sides of this equation to get:
(1400/280)^(1/5) - 1 = r
solve for r to get:
r = .379729661
that's the rate of growth per hour.

replace r in the original equation and solve for f to get:
f = 280 * (1 + .379729661) ^ 5 = 1400.

the rate is good.
after 4 hours, the population will be:
f = 280 * (1 + .379729661) ^ 4 = 1014.691529.

1014.691529 is your solution to question a.
i'm not sure how they want you to round this so i left it as is.

to find how long it takes to grow to 2630, the formula becomes:

2630 = 280 * (1 + .379729661) ^ n
divide both sides of the equation to 280 and simplify the right side of the equation to get:
2630/280 = 1.379729661 ^ n
take the log of both sides of the equation to get:
log(2630/280) = log(1.379729661 ^ n)
since log(x^n) = n*log(x), this becomes:
log(2630/280) = n * log(1.379729661).
divide both sides of the equation by log(1.379729661) to get:
log(2630/280) / log(1.379729661) = n
solve for n to get:
n = 6.958794448.

confirm by replacing n in the original equation and solving for f to get:
f = 280 * 1.379729661 ^ 6.958794448 = 2630.
round to 2 decimal places to get n = 6.96

the equation can be graphed as shown below:



let me know if you have any questions.

theo



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