Exponential problem

Question

The count in a bacteria culture was 200 after 15 minutes and 1300 after 30 minutes. Assuming the count grows exponentially,

What was the initial size of the culture? 

Find the doubling period.

Find the population after 90 minutes

When will the population reach 11000

You may enter the exact value or round to 2 decimal places.

Answer 1

200 = a₀e^15k

1300 = a₀e^30k

6.5 = e^15k

ln 6.5 = 15k

1.8718 ≈ 15k

0.1248 ≈ k

2 = e^kt

2 = e^0.1248t

ln 2 = 0.1248t

0.693 ≈ 0.1248t

5.554 ≈ t

Answer 2

200=pe^15r

1300=pe^30r

13/2= e^30r/e^15r

e^15r=6.5

15r=ln6.5

r=(1/15)ln6.5

p= 200/6.5= 400/13= 30 10/13= initial amount

2 =pe^rt

e^rt = 2/p = 2/400/13 =26/400 =13/200

rt = ln(13/200)

t = ln(13/200)/(1/15)ln6.5

t = (39/40)ln(13/200)/ln6.5 = doubling time

a = (400/13)e^90(1/15)ln6.5

= (400/13)e^6ln6.5 = amount after 90 minutes

11,000 = (400/13)e^(t/15)ln6.5 where t = time when the amount reaches 11,000

e^(t/15)ln6.5 = 13,000/400 = 130/4= 65/2 =32.5

(t/15)ln6.5 = ln(32.5)

t =15ln(32.5)/ln6.5