Population Growth - Bacteria Culture at 300 to 30000 in 4.32 Hrs

In summary, a bacteria culture with 300 bacteria grows at a rate proportional to its size. After 3 hours, there are 9000 bacteria. The expression for the number of bacteria after t hours is 300*e^{1.133732*t}. After 4 hours, there will be 27965.04104 bacteria. The growth rate after 4 hours is 31,705 bacteria/hr. The population will reach 30,000 bacteria after approximately 4.06195 hours.
  • #1
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A bacteria culture starts with 300 bacteria and grows at a rate proportional to its size. After 3 hours, there are 9000 bacteria

A.) Find an expression for the number of bacteria after t hours. for this part, i got [tex]300*e^{1.133732*t}[/tex]

B.) Find the number of bacteria after 4 hours. well using the expression from above and subbing in 4, i get 27965.04104

C.) Find the growth rate after 4 hours. this just means that i need to solve for k right?
p(t) = 300e^{kt}
p(3) = 300e^{3k} = 9000
solved for k and got 1.133% right?

D.) After how many hours will the population reach 30000
well setting the equation from part A equal to 30000 and solved for t and got 4.319991


i know that at least one of these are wrong, but i can't figure out which one. can someone tell me what I am doing wrong?
 
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  • #2
I think Part C is wrong. Isn't the growth rate = dp/dt = kp? (k is constant, by the way, and is equal to 1.133732). So use your answer from part B to get the answer to part C. (Simple as multiplying by k).
 
  • #3
awesome, thanks for the help
 
  • #4
ProBasket said:
A bacteria culture starts with 300 bacteria and grows at a rate proportional to its size. After 3 hours, there are 9000 bacteria

A.) Find an expression for the number of bacteria after t hours. for this part, i got [tex]300*e^{1.133732*t}[/tex]

B.) Find the number of bacteria after 4 hours. well using the expression from above and subbing in 4, i get 27965.04104

C.) Find the growth rate after 4 hours. this just means that i need to solve for k right?
p(t) = 300e^{kt}
p(3) = 300e^{3k} = 9000
solved for k and got 1.133% right?

D.) After how many hours will the population reach 30000
well setting the equation from part A equal to 30000 and solved for t and got 4.319991


i know that at least one of these are wrong, but i can't figure out which one. can someone tell me what I am doing wrong?
From problem statement:
{(dB/dt) = k*B} ⇒ B(t) = B0*exp(k*t)
{B(t=0) = 300} ⇒ B0 = 300
{B(t=3) = 9000} ⇒ 9000 = 300*exp{k*(3)} ⇒ k = (1/3)*Loge{9000/300} = (1.1337325)

ITEM #A:
B(t) = 300*exp{(1.1337325)*t}

ITEM #B:
B(4) = 300*exp{(1.1337325)*(4)} = (27,965 bacteria)

ITEM #C:
(dB/dt)t=4 = k*B(4) = (1.1337325)*(27965) = (31,705 bacteria/hr)

ITEM #D:
(30000) = 300*exp{(1.1337325)*t} ⇒ t = (1.1337325)(-1)*Loge{30000/300}
t = (4.06195 hr)


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1. How does the population of bacteria change over time in this culture?

The population of bacteria in this culture grows exponentially over time, starting at 300 bacteria and reaching 30,000 after 4.32 hours. This rapid growth is due to the bacteria's ability to reproduce quickly in favorable conditions.

2. What factors contribute to the growth of bacteria in this culture?

The growth of bacteria in this culture is influenced by several factors, including the availability of nutrients, temperature, pH level, and oxygen levels. These factors provide an optimal environment for the bacteria to reproduce and thrive.

3. How can we control or limit the growth of bacteria in this culture?

There are several ways to control or limit the growth of bacteria in this culture. One method is to adjust the environmental conditions, such as reducing the nutrient supply or altering the temperature or pH level. Another approach is to introduce antibacterial agents or use sterilization techniques to eliminate or inhibit the growth of bacteria.

4. What is the significance of studying population growth in bacteria culture?

Studying population growth in bacteria culture is important for several reasons. It helps us understand the basic principles of population dynamics and growth rates in microorganisms. This knowledge can also be applied to various fields, such as medicine and agriculture, to control or prevent the spread of harmful bacteria.

5. How does the growth rate of bacteria in this culture compare to other organisms?

The growth rate of bacteria in this culture is generally much faster compared to other organisms due to their small size and ability to reproduce quickly. However, the growth rate can vary depending on the type of bacteria and the environmental conditions. Some bacteria may have slower growth rates, while others can double their population in a matter of minutes.

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