A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 590.

Question

(a) Find an expression for the number of bacteria after t hours

P(t)=______

(b) Find the number of bacteria after 2 hours. (Round your answer to the nearest whole number.)

P(2)=_______ bacteria

(c) Find the rate of growth after 2 hours. (Round your answer to the nearest whole number.)

P'(2)=_______ bacteria per hour

(d) When will the population reach 10,000? (Round your answer to one decimal place.)

t=_______ hr

Answer 1

(1) N = 100. (5.9)^t

(2) 3481

(3) 6161.37

(4) t ~ 2.6 hours

Answer 2

a) Let y be the number of bacterial cells, after time, t. We assume that bacteria grows exponentially

1. y(t) = y(0)e^rt = 100e^rt
2. Since r is the relative growth rate, we must solve for it
3. After 1 hour there are 590 cells so...590 = 100e^r(1)
4. r = ln(5.9)
5. Substitute r into the general equation
6. y(t) = 100e^(ln5.9)t
7. Simplify
8. y(t) = 100*5.9^t

b) After 2 hours, t=2...

1. y(2) = 100*5.9²
2. =3481 cells

c) The growth rate at time "t" is ry(t) = ln(5.9)y(t). After 2 hours...

1. ln(5.9) * 3481 = 6179 cells per hour

d) 1000 = 100*5.9^t

1. t = log_5.9100 = ln(100)/ln(5.9) = 2.6 hours