At time t=0, a bacterial culture weighs 1 gram. Two hours later, the culture weighs 4
grams. The maximum weight of the culture is 20 grams.
(a) Write a logistic equation that models the weight of the bacterial culture.
(b) Find the culture's weight after 5 hours.
(c) When will the culture's weight reach 18 grams?
(d) Write a logistic dierential equation that models the growth rate of the culture's weight. Then
repeat part (b) using Euler's method with a step size of h=1.
(e) After how many hours is the culture's weight increasing most rapidly? Show work.

part (d)= The logistic differential equation used to simulate how quickly a culture's weight increases is [tex]\frac{dy}{dy}= 0.779y(1-\frac{y}{20})[/tex]

5. hours later, the culture weighs 11.571 grams.

part (e)= According to portion (c) of this question, the culture's weight will be 18 grams when t=6.6, which is why the weight of the culture increases most quickly at that time.

what is logistic function?

Given a growth rate, r, and a carrying capacity, K, the logistic equation is a straightforward differential equation model that can be used to connect the change in population, d P d t, to the existing population, P.

given

a bacterial culture weighs 1 gram in time(t)=0

the culture weighs 4 after 2 hours

culture maximum weight is 20 grams.

for part (a)

The general solution for the logistic differential equation is [tex]y(t)=\frac{L}{1+be^{-kt} }[/tex]

The culture can weigh up to 20 grams , so put value of L= 20 grams

The value of [tex]e^{0}[/tex]is 1.

1+b Equals 20 when multiplied on both sides.

on solving b=19

[tex]y(t)= \frac{20}{19e^{-kt} }[/tex]

The culture weights 4 grams at the end of two hours, or y(2)=4.

value of t and y in [tex]y(t)= \frac{20}{19e^{-kt} }[/tex]

on simplifying k≈0.779

b and k values should be substituted in the general solution [tex]y(t)= \frac{20}{19e^{-kt} }[/tex]

Therefore, the model's necessary solution is [tex]y=\frac{20}{1+19e^{0.779t} }[/tex] in logistic equation.

for part (b)

The weight of the culture is 14.425 grams after five hours.

for part (c)

After 6.6 years, the culture has an 18 gram weight.

for part (d)

The logistic differential equation used to simulate how quickly a culture's weight increases is [tex]\frac{dy}{dy}= 0.779y(1-\frac{y}{20})[/tex]

5. hours later, the culture weighs 11.571 grams.

for part (e)

According to portion (c) of this question, the culture's weight will be 18 grams when t=6.6, which is why the weight of the culture increases most quickly at that time.

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