# A Bacteria Culture Initially Contains 100 Cells

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.)

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