A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 310.
- Find an expression for the number of bacteria after t hours.
- Find the number of bacteria after 4 hours. (Round your answer to the nearest whole number.)
- Find the rate of growth after 4 hours. (Round your answer to the nearest whole number.)
- When will the population reach 10,000? (Round your answer to one decimal place.)
Answer:
a) Let y be the number of bacterial cells, after time, t. We assume that bacteria grows exponentially
y(t) = y(0)ert = 100ert
Since r is the relative growth rate, we must solve for it
After 1 hour there are 590 cells so…590 = 100er(1)
r = ln(5.9)
Substitute r into the general equation
y(t) = 100e(ln5.9)t
Simplify
y(t) = 100*5.9t
b) After 2 hours, t=2…
y(2) = 100*5.92
=3481 cells
c) The growth rate at time “t” is ry(t) = ln(5.9)y(t). After 2 hours…
ln(5.9) * 3481 = 6179 cells per hour
d) 1000 = 100*5.9t
t = log5.9100 = ln(100)/ln(5.9) = 2.6 hours
