The number of bacteria in a culture is increasing according to the law of exponential growth. After 5 hours there are 175 bacteria in the culture and after 10 hours there are 425 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 15,000?
A) (i) 72.0588 ; (ii) ; (iii) 3,819.3668 ; (iv) 32.4162 hr
B) (i) 74.2088 ; (ii) ; (iii) 5,194.0840 ; (iv) 34.6442 hr
C) (i) 72.0588 ; (ii) ; (iii) 2,506.6327 ; (iv) 30.0817 hr
D) (i) 77.8388 ; (ii) ; (iii) 7,945.5374 ; (iv) 36.7554 hr
E) (i) 79.3988 ; (ii) ; (iii) 10,598.0009 ; (iv) 38.5348 hr

Question

The number of bacteria in a culture is increasing according to the law of exponential growth. After 5 hours there are 175 bacteria in the culture and after 10 hours there are 425 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 15,000?
A) (i) 72.0588 ; (ii) ; (iii) 3,819.3668 ; (iv) 32.4162 hr 
B) (i) 74.2088 ; (ii) ; (iii) 5,194.0840 ; (iv) 34.6442 hr 
C) (i) 72.0588 ; (ii) ; (iii) 2,506.6327 ; (iv) 30.0817 hr 
D) (i) 77.8388 ; (ii) ; (iii) 7,945.5374 ; (iv) 36.7554 hr 
E) (i) 79.3988 ; (ii) ; (iii) 10,598.0009 ; (iv) 38.5348 hr

Quizplus · Accepted Answer

(i) Using the formula for exponential growth, A = A0e^(kt), we can form two equations:
175 = A0e^(5k)
425 = A0e^(10k)
Dividing the second equation by the first equation, we get:
425/175 = e^(10k)/e^(5k)
2.4286 = e^(5k)
Taking the natural logarithm of both sides, we get:
ln(2.4286) = ln(e^(5k))
ln(2.4286) = 5k
k = 0.2687
Substituting k into the first equation and solving for A0, we get:
175 = A0e^(0.2687*5)
A0 = 72.0588

(ii) Exponential growth model:
A = 72.0588e^(0.2687t)

(iii) After 20 hours:
A = 72.0588e^(0.2687*20)
A ≈ 2,506.6327

(iv) Solving for t when A = 15,000:
15,000 = 72.0588e^(0.2687t)
t ≈ 30.0817 hours

The Number of Bacteria in a Culture Is Increasing According