The number of bacteria in a culture is increasing according to the law of exponential growth.After 5 hours there are 130 bacteria in the culture and after 10 hours there are 330 bacteria in the culture.Answer the following questions,rounding numerical answers to four decimal places.
(I)Find the initial population.

(II)Write an exponential growth model for the bacteria population.Let t represent time in hours.

(III)Use the model to determine the number of bacteria after 20 hours.

(IV)After how many hours will the bacteria count be 10,000?

I)51.2121 ; (I
I) ; (II
I)3,439.1838 ; (I
V)30.6439 hr
I)53.3621 ; (I
I) ; (II
I)4,813.9010 ; (I
V)32.8719 hr
I)51.2121 ; (I
I) ; (II
I)2,126.4497 ; (I
V)28.3094 hr
I)56.9921 ; (I
I) ; (II
I)7,565.3544 ; (I
V)34.9831 hr
I)58.5521 ; (I
I) ; (II
I)10,217.8179 ; (I
V)36.7625 hr

Question

The number of bacteria in a culture is increasing according to the law of exponential growth.After 5 hours there are 130 bacteria in the culture and after 10 hours there are 330 bacteria in the culture.Answer the following questions,rounding numerical answers to four decimal places. ​
(I)Find the initial population.
​
(II)Write an exponential growth model for the bacteria population.Let t represent time in hours.
​
(III)Use the model to determine the number of bacteria after 20 hours.
​
(IV)After how many hours will the bacteria count be 10,000?
​
I)51.2121 ; (I
I) ; (II
I)3,439.1838 ; (I
V)30.6439 hr 
I)53.3621 ; (I
I) ; (II
I)4,813.9010 ; (I
V)32.8719 hr 
I)51.2121 ; (I
I) ; (II
I)2,126.4497 ; (I
V)28.3094 hr 
I)56.9921 ; (I
I) ; (II
I)7,565.3544 ; (I
V)34.9831 hr 
I)58.5521 ; (I
I) ; (II
I)10,217.8179 ; (I
V)36.7625 hr

Quizplus · Accepted Answer

(I) Using the formula for exponential growth, we have:
N(t) = N0 * e^(rt)
where N(t) is the population after time t, N0 is the initial population, r is the growth rate, and e is the base of the natural logarithm. We have two equations:
130 = N0 * e^(5r)
330 = N0 * e^(10r)
Dividing the second equation by the first and simplifying, we get:
e^(5r) = 2.5385
Taking the natural logarithm of both sides, we get:
5r = ln(2.5385) = 0.9310
Solving for r, we get:
r = 0.1862
Substituting r into either equation above and solving for N0, we get:
N0 = 51.2121
Therefore, the initial population is 51.2121.
(II) Using the value of r from above, we have the exponential growth model:
N(t) = 51.2121 * e^(0.1862t)
(III) Using the exponential growth model above, we have:
N(20) = 51.2121 * e^(0.1862 * 20) = 2,126.4497
Therefore, there are approximately 2,126.4497 bacteria after 20 hours.
(IV) Using the exponential growth model above, we need to solve for t when N(t) = 10,000. Setting this up and solving for t, we get:
10,000 = 51.2121 * e^(0.1862t)
e^(0.1862t) = 195.1441
0.1862t = ln(195.1441) = 5.2718
t = 28.3094
Therefore, it will take approximately 28.3094 hours for the bacteria count to reach 10,000.

The Number of Bacteria in a Culture Is Increasing According