Question

A certain type of bacteria in a lab experiment multiplies rapidly. Initially, there are 200 bacteria, and the population triples every 4 hours. The recursive formula for this growth is B(n)=B(n−4)⋅3
𝐵
(
𝑛
)
=
𝐵
(
𝑛
−
4
)
⋅
3
, where B(n)
𝐵
(
𝑛
)
is the number of bacteria after n hours, and B(0)=200
𝐵
(
0
)
=
200
. If the experiment continues without interruption, how many hours will it take for the bacteria population to exceed 2,700,000?

Answer 1

To find how many hours it will take for the bacteria population to exceed 2,700,000, we start with the given recursive formula:

\[
B(n) = B(n-4) \cdot 3
\]

and initial condition:

\[
B(0) = 200
\]

This implies:
- After 4 hours:
\[
B(4) = B(0) \cdot 3 = 200 \cdot 3 = 600
\]
- After 8 hours:
\[
B(8) = B(4) \cdot 3 = 600 \cdot 3 = 1800
\]
- After 12 hours:
\[
B(12) = B(8) \cdot 3 = 1800 \cdot 3 = 5400
\]
- After 16 hours:
\[
B(16) = B(12) \cdot 3 = 5400 \cdot 3 = 16200
\]
- After 20 hours:
\[
B(20) = B(16) \cdot 3 = 16200 \cdot 3 = 48600
\]
- After 24 hours:
\[
B(24) = B(20) \cdot 3 = 48600 \cdot 3 = 145800
\]
- After 28 hours:
\[
B(28) = B(24) \cdot 3 = 145800 \cdot 3 = 437400
\]
- After 32 hours:
\[
B(32) = B(28) \cdot 3 = 437400 \cdot 3 = 1312200
\]
- After 36 hours:
\[
B(36) = B(32) \cdot 3 = 1312200 \cdot 3 = 3936600
\]

Now we summarize the bacterial population at these intervals:
- \(B(0) = 200\)
- \(B(4) = 600\)
- \(B(8) = 1800\)
- \(B(12) = 5400\)
- \(B(16) = 16200\)
- \(B(20) = 48600\)
- \(B(24) = 145800\)
- \(B(28) = 437400\)
- \(B(32) = 1312200\)
- \(B(36) = 3936600\)

We need the number of bacteria to exceed 2,700,000. From the above calculations, we can see:

- At 32 hours, the population is 1,312,200.
- At 36 hours, the population is 3,936,600.

Therefore, the experiment will take **36 hours for the bacteria population to exceed 2,700,000.**