The count in a culture of bacteria was 600 after 2 hours and 38,400 after 6 hours. (a) What is the relative rate of growth of the bacteria population? Express your answer as a percentage. (Round your answer to the nearest whole number.) % (b) What was the initial size of the culture? (Round your answer to the nearest whole number.) bacteria (c) Find a function that models the number of bacteria n (t) after t hours. (Enter your answer in the form n0ert. Round your n0 value to the nearest whole number. Round your r value to two decimal places.) n (t) = (d) Find the number of bacteria after 4.5 hours. (Round your answer to the nearest hundred.) bacteria (e) After how many hours will the number of bacteria reach 75,000? (Round your answer to two decimal places.) hr

A) 2033%

B) 75 bacteria

C) n (t) = 75 (1 + 1.83%) ^t R = 1.83%

D) 81.4 bacteria

E) 348.83 seconds

Step-by-step explanation:

Given that the count in a culture of bacteria was 600 after 2 hours and 38,400 after 6

A) growth (600 - 0) / 2 = 300 bacteria/s

38400/6 = 6400 bacteria/s

Relative growth = (6400-300) / 300 * 100

= 6100/300 * 100

= 2033%

B) initial size of the culture

Using exponential equation

P = I (1 + R) ^t

Where I = initial size

R = rate

600 = I (1 + R) ^2

Log both sides

Log600 = log I (1+R) ^2

2.778 = logI + 2log (1 + R)

2log (1+R) = 2.778 - logI ... (1)

Also,

38400 = I (1 + R) ^6

Log both sides

Log 38400 = logI + 6log (1+R)

6Log (1+R) = 4.584 - logI ... (2)

Divide equation 2 by 1

6/2 = (4.584 - logI) / (2.778 - logI)

Cross multiply

16.668 - 6logI = 9.168 - 2logI

6logI - 2logI = 16.668 - 9.168

4logI = 7.5

LogI = 7.5/4

LogI = 1.875

I = 74.98 = 75 bacteria

C) A function that models the number of bacteria n (t) after t hours.

If I = 75 bacteria

Then n (t) = 75 (1 + R) ^t

600 = 75 (1+R) ^2

8 = (1+R) ^2

Log both sides

Log8 = 2log (1+R)

0.903/2 = log (1+R)

0.45 = log (1+R)

1 + R = 2.83

R = 1.83%

The model function is therefore

n (t) = 75 (1 + 1.83%) ^t

D) the number of bacteria after 4.5 hours

n (t) = 75 (1.02) ^4.5

n (t) = 81.4 bacteria

E) After how many hours will the number of bacteria reach 75,000

n (t) = 75 (1.02) ^t

75000 = 75 (1.02) ^t

1000 = 1.02^t

Log both sides

Log 1000 = tlog 1.02

3 = 0.0086t

t = 348.83 seconds