# Growth Problems

• Nov 12th 2012, 12:26 AM
camjenson
Growth Problems
Colony of bacteria grows at rate directly proportional to the size of the colony. The colony triples every two hours.
1. Write the differential equation describing the growth and the general solution to the equation.
2. What's the value of constant of proportionality?
3. At t=9 hours, the pop is 701.44 million, what is the initial population?
• Nov 12th 2012, 12:39 AM
Prove It
Re: Growth Problems
What have you tried?
• Nov 12th 2012, 12:44 AM
fkf
Re: Growth Problems
Let us call the size of the colony y(t), where y is a function of the time t. Therefore the size of the colony depends on the time. Then call the rate of growth y'(t), were t is the time again. We're given that the rate of growth is proportional to the size of the colony, call this proportional factor k. Hence we have

y'(t) = k*y(t) <=> y'(t) - k*y(t) = 0
Solving this differential equation gives us the general solution
y(t) = C*e^(kt)

Use the general solution to solve the value of k using the fact the the colony triples every third hour. We also know that when t = 0 we have
y(0) = C*e^(k0) = C
This means that C is the initial value of the problem which in this case is the population of the bacteria colony.