Exponential growth and decay

Question

Kelsey is culturing bacteria that have a growth rate of 10% per hour. If the current population is 34,000 bacteria, how many bacteria will there be in 16 hours?

If necessary, round your answer to the nearest whole number.

Answer 1

A(t) = A₀e^kt

 

A₀ = 34000

 

We need to find k.  Note that after the first hour, A goes from A₀ to A₀ + (1/10)A₀ = 1.1A₀

 

A(1) = A₀e^k(1) = 1.1A₀ ⇒ 

 

e^k = 1.1 ⇒

 

k = (ln1.1) hours^-1

 

A(t) = 34000e^(ln1.1)t, with t in hours

 

A(16) = 34000e^(ln1.1)(16) = 156229

 

Note:  Michael's solution gives 34000(1.10)¹⁶ = 156229 = same answer.  The compound interest formula works here too.

Answer 2

Exponential Growth

A(t) = a₁e^kt

A(16) = 34,000e^(1.1)(16)

A(16) = 34000e^17.6

A(16) = 34000(44013193)

Answer 3

For exponential growth,

 

P(t) = P_o(1 + r)^t

 

where:
P = amount after certain time

P_o =current amount

r = rate (as decimal)

t = time

 

t=0  is the time of current amount of bacteria.

 

The formula is then

 

P(t)= 34000(1.10)^t

 

Using this formula, evaluate P(16).