The rate of growth of a certain cell culture is proportional to its size. in 10 hours a population of 1 million cells grew to 99 million. how large will the cell culture be after 25 hours

The number of cells in the culture after 25 hours is  9.34 × 10^25 cells.

According to the question, rate of growth of a certain cell culture is proportional to its size

Let the rate of growth of the cells be dP/dt

Let the time taken be t

Let P be the number cells in the culture

According to the question;

dP/dt α P

Introducing the constant of proportionality k

dP/dt = kP

Collecting like terms

dP/P = kdt

Integrating both sides

lnPt - lnPo + C = kt + C

Eliminating the constants of integration

lnPt - lnPo = kt

We can obtain k from

lnPt - lnPo = kt

Pt = population at time = t

Po = population at time = 0

When

Po = 1 × 10^6

Pt = 99 × 10^6

t = 10 hours

k = lnPt - lnPo/t

k = ln (99 × 10^6 -  1 × 10^6)/10

k = 1.84 cells hour-1

To obtain Pt when t = 25 hours;

lnPt = kt + lnPo

Pt = e^(1.84 cells hour-1  × 25 hours) +  ln(1 × 10^6)

Pt = 9.34 × 10^25 cells

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98 billion

Explanation:

1. Expression for rate of growth

Let Po = the original population

and P = the population at time t. Then

dP/dt = kP

 dP/P = kdt

   lnP = kt + C

At t = 0, lnPo = C

      lnP = kt + lnPo

ln(P/Po) = kt

2. Value of k

ln(99 million/1 million) = k(10)

ln99 = 10k

k = ln99/10 = 4.595/10 = 0.4595/h

ln(P/Po)  = 0.4595t

3. Population after 25 h

ln(P/Po) = 0.4595× 25 = 11.49

    P/Po = e^(11.49) = 97 519

          P = 97519No = 97 519 × 1 × 10^6 = 9.8 × 10^10 = 98 × 10^9

          P = 98 billion.

In 25 h the population of the cell culture will be 98 billion.