A bacterial culture starts with 500 bacteria and doubles in size every half-hour.(a) How many bacteria are there after 3 hours?
Answer: Since 3 hours equals 6 half-hours, the culture will have doubled 6 times.
Therefore, there will be
500 · 2
6 = 32,000
bacteria.
(b) How many bacteria are there after t hours?
Answer: Since t hours is the same as 2t half-hours, the culture will have doubled 2t
times. Therefore, there will be
500 · 2
2t
bacteria.
(c) How many bacteria are there after 40 minutes?
Answer: There are two possible answers depending on how you interpret the set-up
to the problem. If each bacterium in the culture doubles once every half-hour on the
half-hour, then each one will double after exactly 30 minutes, and then not again until
60 minutes have passed. In that case, there will be
500 · 2 = 1000
4
bacteria after 40 minutes.
On the other hand, if each bacterium doubles exactly once per half-hour, but at some
random time within that half-hour, then it makes sense to think of the population
function P(t) = 500 · 2
2t as continuous. In that case, since 40 minutes is
40
60
=
2
3
of an hour, the population will be
500 · 2
2
2
3 = 500 · 2
4
3 ≈ 1259
after 40 minutes.
Answers
Answer:
a). 32000
b).
c). 1259
Step-by-step explanation:
Growth of a bacteria is always exponential. Therefore, population of the bacteria is represented by the the geometric sequence.
Sum of the bacterial population after t hours will be represented by
Where a = population at the start
r = ratio with the population is growing
n = time or duration of the growth in one hour
a). Population of 500 bacteria gets doubled after half an hour.
Or gets 4 times after an hour
This sequence will have a common ratio r = 4
and initial population a = 500
Therefore, population of the bacteria after 3 hours will be
b). After t hours number of bacteria will be represented by
c). We have to calculate the population after 40 minutes.
That means duration 't' = 40 minutes of hours
By the formula,
≈ 1259
Therefore, number of bacteria after 40 minutes will be 1259.
After 3 hours, there will be 32,000 bacteria in the culture, given that the bacteria double in size every half-hour. The number of bacteria at any given time depends on whether they double precisely every half-hour or continuously within that timeframe.
In this scenario, the population growth of the bacterial culture follows exponential growth, where it doubles every half-hour. To calculate the number of bacteria after 3 hours (equivalent to 6 half-hours), you can use the formula for exponential growth: P(t) = P₀ * , where P(t) is the population at time t, P₀ is the initial population, t is the time in hours, and h is the time interval for doubling (in this case, 0.5 hours). Plugging in the values, you get P(3) = 500 *
= 32,000 bacteria.
This means that after 3 hours, there will be 32,000 bacteria in the culture. The explanation also addresses the alternate interpretation of continuous growth, where the population increases continuously within each half-hour, resulting in approximately 1259 bacteria after 40 minutes.
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Answer:
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Step-by-step explanation:
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