Answer:
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.
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
[tex]\int\limits^P_0 {\frac{dP}{P} } \, dP = \int\limits^t_0 {k} \, dt[/tex]
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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