The number of a certain type of bacteria increases continuously at a rate proportional to the number present.There are 100 present initially,and 200 present 6 hours later.How many will there be 20 hours after the initial time? Round your answer to the nearest integer.
A) 10 bacteria
B) 903 bacteria
C) 1,008 bacteria
D) 23 bacteria
E) 18 bacteria

Question

​The number of a certain type of bacteria increases continuously at a rate proportional to the number present.There are 100 present initially,and 200 present 6 hours later.How many will there be 20 hours after the initial time? Round your answer to the nearest integer. ​
A) ​10 bacteria
B) 903 ​bacteria
C) ​1,008 bacteria
D) ​23 bacteria
E) ​18 bacteria

Quizplus · Accepted Answer

Let the number of bacteria be denoted by N(t).

According to the problem, the rate of increase of bacteria is proportional to the number present i.e,

dN/dt = kN

where k is the proportionality constant.

Solving the differential equation, we get:

N(t) = N₀e^(kt)

Given that N₀ = 100 and N(6) = 200, we can find k as follows:

200 = 100e^(6k)
=> e^(6k) = 2
=> 6k = ln2
=> k = (ln2)/6

Therefore, N(20) = 100e^((ln2/6) * 20) ≈ 1008, rounded to the nearest integer.

Hence, the best choice is C.

​The Number of a Certain Type of Bacteria Increases Continuously

The Number of a Certain Type of Bacteria Increases Continuously