A bacteria culture is known to grow at a rate proportional t

A bacteria culture is known to grow at a rate proportional to the amount present. There are 10 strands of bacteria after 4 hour(s) in the culture; and 20 strands after 7 hours. a. Find the amount of bacterias N as function of time. N(t) =

Solution

Let dx/dt = the rate of growth of the bacteria
Let t = time (in hours) after start of culture
Let x = amount of bacteria present
Let k = constant of proportionality

dx/dt = kx
dx/x = k dt
ln x = kt + C <--- integrate both sides
x = e^(kt + C) <--- raise both sides to e
x = e^kt * e^C
x = e^kt * B <--- e^C , a constant raised to a constant, is a constant, so we represent it with a new constant, B

x = Be^kt

When t = 4, x = 10
10 = Be^(k*4)
10 = Be^(4k)
10/(e^(4k)) = B

When t = 7, N = 20. We substitute 10/(e^(4k)) = B
20 = 10/(e^(4k)) * e^(7k)
2 = (e^3k)
³2 = e^k

10/(³2) = B

N(t) = B * (³2) ^ t
N(t) = 10/(³2) * (³2) ^ t
N(t) = 1000 (³2) ^ (t - 1)