The number of bacteria in a culture is modeled by:
n(t) = 1330e^(0.42t)
(a) The initial number of bacteria is:
(b) The relative rate of growth of this bacterium population is:
(c) The number of bacteria after 3 hours is:
(d) After how many hours will the number of bacteria reach 10,000?
Your answer is:
To find the answers to these questions, we can use the equation provided: n(t) = 1330e^(0.42t).
(a) The initial number of bacteria is found by plugging in t = 0 into the equation:
n(0) = 1330e^(0.42 * 0)
= 1330e^0
= 1330 * 1
= 1330
So, the initial number of bacteria is 1330.
(b) The relative rate of growth of the bacterium population can be found by taking the derivative of the equation with respect to time (t):
n'(t) = 1330 * 0.42 * e^(0.42t)
= 559.8 * e^(0.42t)
So, the relative rate of growth of this bacterium population is 559.8 * e^(0.42t).
(c) The number of bacteria after 3 hours can be found by substituting t = 3 into the equation:
n(3) = 1330e^(0.42 * 3)
= 1330 * e^1.26
= 1330 * 3.524368
= 4683.0708
So, the number of bacteria after 3 hours is approximately 4683.0708.
(d) To find out after how many hours the number of bacteria will reach 10,000, we need to solve the equation n(t) = 10000 for t.
So, we have:
1330e^(0.42t) = 10000
Dividing both sides of the equation by 1330:
e^(0.42t) = 7.5188
Taking the natural logarithm of both sides:
0.42t = ln(7.5188)
Dividing both sides of the equation by 0.42:
t = ln(7.5188) / 0.42
Using a calculator, we can find the value of ln(7.5188) / 0.42 to be approximately 4.375.
So, after approximately 4.375 hours, the number of bacteria will reach 10,000.
Final answers:
(a) The initial number of bacteria is 1330.
(b) The relative rate of growth of this bacterium population is 559.8 * e^(0.42t).
(c) The number of bacteria after 3 hours is approximately 4683.0708.
(d) After approximately 4.375 hours, the number of bacteria will reach 10,000.