PLEASE HELP QUICK ASAP

Question

Bacteria are a type of biological cell. A certain bacterium doubles every hour. The initial population of a culture is 130 cells.

a. Find a formula for the bacteria population after 𝑛 hours.

b. Determine the population after 4 hours.

c. Determine the length of time (to the nearest minute) required for the population to

Answer 1

doubling time in 1 year, at continuously compound growth means

260=130e^r(1)

2= e^r

take natural logs of both sides

ln2 = r

r = ln2 = about .69

A= 130e^.69n or 130e^(nln2)

After 4 hours it doubled 4 times

from 130 to 260,

then to 520 after 2 hours

then to 1040 after 3 hours,

then to 2080 after 4 hours

or

A = 130e^4ln2 = 130(16) = 2080 cells

to determine time in minutes when it reaches a given level of cells,

set A= to the number of cells and solve for m

A= 130e^(m/60)ln2

A/130 = e^(m/60)ln2

take natural logs of both sides

ln(A/130) = (m/60)ln2

m/60 = ln(A/130)/ln2

m= 60ln(A/130)/ln2

or just solve for t and use

A=130e^tln2 where t= hours,

ln(A/130)= tln2

t = ln(A/130)/ln2 = number of hours to reach a given level, A, of cells = log₂(A/130)

then multiply by 60 , 60t, to get the number of minutes:

m=60t = numeber of minutes to reach A, = 60t = 60ln(A/130)/ln2 or 60log₂(A/130)

Answer 2

Your third question is cut off.

Lets start with the population doubles every hour. If we represent initial population as P, after 1 hour, the population will be 2xP. After another hour, the population doubles again and will be 2x2xP. After three hours from the start, the population will have doubled again, 2x2x2xP. Therefore, the population at any time, n, can be represented as: Population=Px2ⁿ

1. Population = 130 x 2ⁿ

2. Just substitute 4 for n and solve

Population(4) = 130 x 2⁴ = 2080

3. I assume question 3 asks you to solve for the time when the population reaches a certain value. We can easily solve for this by dividing the 130 from both sides and taking the log₂ of both sides.

n = log₂(population /130)

This will result in a number of hours. To convert to hours and minutes, take everything to the right of the decimal point and multiply by 60