The number of bacteria in a culture is modeled by
n(t)=1710e071t
(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
I got the A=1710
B=?
c=14389
D=?
I keep getting b and d wrong any suggestions. thanks in advance!
a. ok.
b.
c. ok.
d. 1710*e^0.71t = 10,000
e^0.71t = 10,000 / 1710 = 5.85
0.71t*Lne = Ln5.85
Divide both sides by Lne:
0.71t = Ln5.85 / Lne = 1.7664
t = 1.7664 / 0.71 = 2.49 Hours.
To find the answers to parts (a) and (b) of the question, we can look at the given function for the number of bacteria in the culture: n(t) = 1710e^(0.71t).
(a) The initial number of bacteria is the value of n(t) at t = 0. To find this, we substitute t = 0 into the function:
n(0) = 1710e^(0.71 * 0) = 1710e^0 = 1710 * 1 = 1710.
So, the initial number of bacteria is 1710.
(b) The relative rate of growth of a population can be determined by taking the derivative of the function n(t) with respect to t. Let's find dn/dt:
dn/dt = d/dt (1710e^(0.71t)) = 1710 * 0.71 * e^(0.71t).
The relative rate of growth of the bacteria population is the expression 1710 * 0.71 * e^(0.71t). Note that this value will change at different points in time (t).
Moving on to parts (c) and (d) of the question:
(c) To find the number of bacteria after 3 hours, we can substitute t = 3 into the function:
n(3) = 1710e^(0.71 * 3) ≈ 1710e^(2.13) ≈ 1710 * 8.435 ≈ 14,430.85.
So, the number of bacteria after 3 hours is approximately 14,431.
(d) To determine how many hours it takes for the number of bacteria to reach 10,000, we can set n(t) equal to 10,000 and solve for t:
10,000 = 1710e^(0.71t).
To solve this equation, we can first divide both sides by 1710:
10,000/1710 = e^(0.71t).
Next, take the natural logarithm (ln) of both sides:
ln(10,000/1710) = 0.71t.
Finally, divide both sides by 0.71 to solve for t:
t = ln(10,000/1710) / 0.71 ≈ 2.94 hours.
So, the number of bacteria will reach 10,000 after approximately 2.94 hours.
To recap:
(a) The initial number of bacteria is 1710.
(b) The relative rate of growth of this bacterium population is 1710 * 0.71 * e^(0.71t).
(c) The number of bacteria after 3 hours is approximately 14,431.
(d) The number of bacteria will reach 10,000 after approximately 2.94 hours.