The amount of bacteria present in a culture at time t in hours obeys the function
N(t) = 900e^.13t
a) Determine the number of bacteria at t = 0 hours?
b) What is the growth rate of the bacteria?
c) What is the population after 4 hours?
d) When will the population reach 2500?
e) How long will it take to double?
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a) To determine the number of bacteria at t = 0 hours, we substitute t = 0 into the function N(t):
N(0) = 900e^0 = 900 * 1 = 900
Therefore, the number of bacteria at t = 0 hours is 900.
b) The growth rate of the bacteria can be found by considering the coefficient before the exponential term. In this case, the coefficient is 0.13. Therefore, the growth rate of the bacteria is 0.13.
c) To find the population after 4 hours, we substitute t = 4 into the function N(t):
N(4) = 900e^(0.13*4) = 900e^(0.52) ≈ 900 * 1.68 ≈ 1512
Therefore, the population after 4 hours is approximately 1512.
d) To find when the population will reach 2500, we can set the function N(t) equal to 2500 and solve for t:
2500 = 900e^(0.13t)
Dividing both sides by 900:
2.7778 = e^(0.13t)
Taking the natural logarithm of both sides:
ln(2.7778) = 0.13t
Dividing both sides by 0.13:
t ≈ ln(2.7778)/0.13 ≈ 7
Therefore, the population will reach 2500 after approximately 7 hours.
e) To find how long it will take to double the population, we set N(t) equal to twice the initial population (2 * 900 = 1800) and solve for t:
1800 = 900e^(0.13t)
Dividing both sides by 900:
2 = e^(0.13t)
Taking the natural logarithm of both sides:
ln(2) = 0.13t
Dividing both sides by 0.13:
t ≈ ln(2)/0.13 ≈ 5.35
Therefore, it will take approximately 5.35 hours to double the population.
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