# Math

(Growth of Cells) Suppose that after t hrs, there are p (t) cells present in a culture, where P(t) = 5000e*^2t

a) How many cells were present initially?
b) Give a differential equation satisfied by P(t)
c) When will the population double?
d) When will 20,000 cells be present?

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1. a. "initially" means time = 0
P(0) = 5000e^2(0) = 5000e^0 = 5000

b. P(t) = 5000*e^(2t)

[d/du]e^u = e^u du

Therefore, P'(t) = 5000*2e^(2t)
P'(t) = 10000e^(2t)

c. Set e^(2t) = 2.
2t = ln2
t = (ln2)/2

The population will double every (ln2)/2 hours.

d. set P = 20000 and solve for t.

20000 = 5000e^(2t)
4 = e^(2t)
ln4 = 2t
2ln2 = 2t
ln2 = t

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