Question

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.

Answer 1

Answer:

a). 32000

b). $T_{t}=500\times 4^{t}$

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

$T_{n}=ar^{n}$

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

$T_{3}=500\times 4^{3}$

$T_{3}=32000$  

b). After t hours number of bacteria will be represented by

$T_{t}=500\times 4^{t}$

c). We have to calculate the population after 40 minutes.

That means duration 't' = 40 minutes of $\frac{2}{3}$ hours

By the formula,

$T_{\frac{2}{3}}=500\times 4^{\frac{2}{3}}$

$T_{\frac{2}{3}}=1259.92$ ≈ 1259

Therefore, number of bacteria after 40 minutes will be 1259.

Answer 2

Answer:

Step-by-step explanation:

We would apply the formula,

y = ab^t

Where

A represents the initial amount of bacteria.

t represents the doubling time.

From the information given

A = 500

t = 1/2 hours

Since after 1/2 hours, the population doubles, then

y = 2 × 500 = 1000

Therefore

1000 = 500 × b^1/2

2 = b^1/2

Raising both sides of the equation by 2, it becomes

2^2 = b^1/2 × 2

b = 4

The equation becomes

y = 500 × 4^t

Therefore, the population in 3 years time would be

y = 500 × 4^3

y = 32000

b) The number of bacteria after t years is

y = 500 × 4^t

c) To determine the number of bacteria after 40 minutes, we would first convert 40 minutes to hours. It becomes

40/60 = 2/3 hours. Therefore

y = 500 × 4^2/3

y = 1259