precalc question

Question

At the beginning of a laboratory experiment, a technician observes 16 CFUs (colony-forming units) of a certain bacterium in a culture. After 4.2 hours, there are 44280 CFUs in the culture. Assuming that the culture is growing exponentially and continues to grow at the same rate, how many CFUs will there be in the culture 6.5 hours after the beginning of the experiment?

Round your answer to the nearest whole number.

Answer:  CFUs

Answer 1

44280/16=e^4.2r

11070/4= 2767.5 =e^4.2r

r= ln2767.5/4.2= about 1.887071349=rate of growth

in 4.2 hours there increase from 16 to 44280

in 6.5 hours how many?

A=16e^4.2(1.887071349) = 44280.00002

A = 16e^1.887071329(6.5)

= 16e^12.2655

=about 3,397,507 CFUs after 6.5 hours

Answer 2

"Assuming that the culture is growing exponentially" means that you can model the population using:

y = ab^x or, in this case, if we let "P" be the bacteria population of CFUs and "t" be the time in hours then:

P = ab^t

At the beginning means at t = 0 so:

P = ab^t plug in t = 0 and P = 16

16 = ab⁰ but since b⁰ = 1 then a = 16

Now we have P = 16b^t now plug in t = 4.2 and P = 44280:

44280 = 16b^4.2

44280/16 = b^4.2

2767.5 = b^4.2

4.2 = 4 2/10 = 4 1/5 = 21/5

2767.5 = b^21/5

(2767.5)^5/21 = (b^21/5)^5/21

6.600011218 = b

Now, the equation is P = 16(6.600011218)^t

Let t = 6.5:

P = 16(6.600011218)^6.5

P = 3397507 CFUs

Answer 3

Exponential model.

y(t) = ab^t

y(0) = a = 16

y(t)=16b^t

y(4.2) = 16b^4.2 = 44280

4.2ln(b) = ln(44280/16) = ln(2767.5)

ln(b) = 7.9257/4.2 = 1.8871

b = e^1.8871 = 6.6

y(t)=16(6.6)^t

y(6.5) = 16(6.6)^6.5 = 16(212344.1882) ≈ 33975507 CFUs