Question

a bacteria culture is known to grow at a rate proportional to the amount present. After one hour, 1000 strands of the bacteria are observed in the culture.and after four hours, 3000 strands. Find a) an expression for the approximate number of strands of the bacteria present in the culture at any time t b) the approximate number of strands of the bacteria originally in the culture.

Answer 1

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 = 1, x = 1000 
1000 = Be^(k*1) 
1000 = Be^k 
1000/(e^k) = B 

When t = 4, x = 3000. We substitute 1000/(e^k) = B 
3000 = 1000/(e^k) * e^(k*4) 
3 = (e^3k) 
³√3 = e^k 

1000/(³√3) = B 

x = B * (³√3) ^ t 
x = 1000/(³√3) * (³√3) ^ t 
x = 1000 (³√3) ^ (t - 1) 

b) 

When t = 0, 
x = 1000 (³√3) ^ (0 - 1) 
x = 693 strands of bacteria

Comments

This helps me a lot