Question

A bacteria culture grows with constant relative growth rate. The bacteria count was 1,920 after 4 hours and 1,966,080 after 12 hours.
(a) What is the relative growth rate? Express your answer as a percentage. (Round your answer to two decimal places.)

(b) What was the initial size of the culture?

(c)Find an expression for the exact number of bacteria after t hours, y(t).

(d) Find the number of bacteria after 9.5 hours. (Round your answer to the nearest whole number.)

(e) Find the rate of growth (in bacteria per hour) after 9.5 hours. (Round your answer to the nearest whole number.)

(f) How many hours did it take for the population to reach 72,000? (Round your answer to two decimal places.)

Answer 1

From the calculation, the growth rate is 0.88.

What is the growth rate?

To find the relative growth rate

P1 = Poe^rt

P2 = Poe^rt

Thus;

1920 = Poe^4r ------ (1)

1966080 =Poe^12r -------(2)

P2/P1 gives;

1966080/1920 = Poe^12r/Poe^4r

1024 = e^12r/e^4r

1024 =e^8r

1024 = (e^r)^8

2^10 = (e^r)^8

e^r = 2^1.25

e^r = 2.38

r = ln(2.38)

r = 0.88

The initial size of the culture is;

1920 = Poe^(0.88 * 4)

Po = 1920/e^(0.88 * 4)

Po = 57

The  expression for the exact number of bacteria after t hours is

P(t) = 57e^0.08t

The growth (in bacteria per hour) after 9.5 hours is

P(t) = 57e^(0.88 * 9.5)

P(t) = 243544

For the number to reach 72,000;

72,000 =  57e^0.88t

72,000/57 = e^0.88t

ln 126 = ln[e^0.88t]

4.8 = 0.88t

t = 4.8/0.88

t = 5.5 hours

Learn more about exponential growth;brainly.com/question/13674608

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