A Bacteria Population Decreases By A Factor Of 1 2 \frac{1}{2} 2 1 ​ Every 4 Hours. Explain Why We Could Also Say That The Population Decays By A Factor Of 1 2 4 \sqrt[4]{\frac{1}{2}} 4 2 1 ​ ​ Every Hour.

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Introduction

The study of population dynamics is a crucial aspect of biology, and understanding the factors that influence population growth or decline is essential for making accurate predictions and informed decisions. In this article, we will explore the concept of population decay and examine why a bacteria population that decreases by a factor of $\frac{1}{2}$ every 4 hours can also be described as decaying by a factor of $\sqrt[4]{\frac{1}{2}}$ every hour.

The concept of population decay

Population decay refers to the reduction in the size of a population over time. This can occur due to various factors, such as environmental changes, predation, disease, or other external influences. In the context of bacteria populations, decay can be caused by factors such as the depletion of nutrients, the presence of antibiotics, or the action of predators.

The given decay rate

We are given that a bacteria population decreases by a factor of $\frac{1}{2}$ every 4 hours. This means that if the initial population size is $P_0$, the population size after 4 hours will be $\frac{1}{2}P_0$, after 8 hours will be $\frac{1}{2}\cdot\frac{1}{2}P_0 = \frac{1}{4}P_0$, and so on.

Understanding the relationship between decay rates

To understand why we can also say that the population decays by a factor of $\sqrt[4]{\frac{1}{2}}$ every hour, let's examine the relationship between decay rates. If a population decreases by a factor of $\frac{1}{2}$ every 4 hours, we can express this as a decay rate of $\left(\frac{1}{2}\right)^{\frac{1}{4}}$ per hour.

Deriving the decay rate

To derive the decay rate, we can use the concept of exponential decay. Exponential decay is a process in which the size of a population decreases at a constant rate over time. The formula for exponential decay is given by:

$$P(t) = P_0e^{-kt}$$

where $P(t)$ is the population size at time $t$, $P_0$ is the initial population size, $k$ is the decay rate, and $t$ is time.

Applying the decay rate formula

In this case, we are given that the population decreases by a factor of $\frac{1}{2}$ every 4 hours. We can express this as a decay rate of $\left(\frac{1}{2}\right)^{\frac{1}{4}}$ per hour. Using the decay rate formula, we can write:

$$P(t) = P_0e^{-\left(\frac{1}{2}\right)^{\frac{1}{4}}t}$$

Simplifying the expression

To simplify the expression, we can use the fact that $e^x = (e^x)^{\frac{1}{4}}$. Therefore, we can rewrite the expression as:

$$P(t) = P_0\left(e^{-\left(\frac{1}{2}\right)^{\frac{1}{4}}}\right)^t$$

Understanding the decay rate in terms of the fourth root

Now, let's examine the decay rate in terms of the fourth root. We can rewrite the expression as:

$$P(t) = P_0\left(\left(\frac{1}{2}\right)^{\frac{1}{4}}\right)^t$$

Simplifying the expression

To simplify the expression, we can use the fact that $\left(a^b\right)^c = a^{bc}$. Therefore, we can rewrite the expression as:

$$P(t) = P_0\left(\frac{1}{2}\right)^{\frac{t}{4}}$$

Understanding the relationship between the decay rates

Now, let's examine the relationship between the decay rates. We can see that the decay rate of $\left(\frac{1}{2}\right)^{\frac{1}{4}}$ per hour is equivalent to a decay rate of $\frac{1}{2}$ every 4 hours.

Conclusion

In conclusion, we have shown that a bacteria population that decreases by a factor of $\frac{1}{2}$ every 4 hours can also be described as decaying by a factor of $\sqrt[4]{\frac{1}{2}}$ every hour. This is because the decay rate of $\left(\frac{1}{2}\right)^{\frac{1}{4}}$ per hour is equivalent to a decay rate of $\frac{1}{2}$ every 4 hours. Understanding the relationship between decay rates is essential for making accurate predictions and informed decisions in the context of population dynamics.

References

• [1] Population dynamics. Encyclopedia Britannica.
• [2] Exponential decay. Wolfram MathWorld.
• [3] Bacteria population dynamics. Journal of Bacteriology.

Further reading

• [1] Population ecology. Oxford University Press.
• [2] Mathematical modeling of population dynamics. Springer.
• [3] Bacteria population modeling. Journal of Mathematical Biology.
  

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Introduction

In our previous article, we explored the concept of population decay and examined why a bacteria population that decreases by a factor of $\frac{1}{2}$ every 4 hours can also be described as decaying by a factor of $\sqrt[4]{\frac{1}{2}}$ every hour. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the relationship between the decay rate and the time period?

A: The decay rate is inversely proportional to the time period. In this case, the decay rate of $\left(\frac{1}{2}\right)^{\frac{1}{4}}$ per hour is equivalent to a decay rate of $\frac{1}{2}$ every 4 hours.

Q: How does the decay rate affect the population size?

A: The decay rate determines the rate at which the population size decreases over time. In this case, the decay rate of $\left(\frac{1}{2}\right)^{\frac{1}{4}}$ per hour means that the population size will decrease by a factor of $\sqrt[4]{\frac{1}{2}}$ every hour.

Q: Can the decay rate be expressed in terms of the exponential function?

A: Yes, the decay rate can be expressed in terms of the exponential function. The formula for exponential decay is given by:

$$P(t) = P_0e^{-kt}$$

where $P(t)$ is the population size at time $t$, $P_0$ is the initial population size, $k$ is the decay rate, and $t$ is time.

Q: How does the decay rate affect the population growth or decline?

A: The decay rate determines the rate at which the population size grows or declines over time. In this case, the decay rate of $\left(\frac{1}{2}\right)^{\frac{1}{4}}$ per hour means that the population size will decline by a factor of $\sqrt[4]{\frac{1}{2}}$ every hour.

Q: Can the decay rate be expressed in terms of the fourth root?

A: Yes, the decay rate can be expressed in terms of the fourth root. The decay rate of $\left(\frac{1}{2}\right)^{\frac{1}{4}}$ per hour is equivalent to a decay rate of $\frac{1}{2}$ every 4 hours.

Q: How does the decay rate affect the population size in the long term?

A: The decay rate determines the rate at which the population size decreases over time. In the long term, the population size will approach zero if the decay rate is greater than zero.

Q: Can the decay rate be expressed in terms of the half-life?

A: Yes, the decay rate can be expressed in terms of the half-life. The half-life is the time it takes for the population size to decrease by half. In this case, the half-life is 4 hours, and the decay rate is $\left(\frac{1}{2}\right)^{\frac{1}{4}}$ per hour.

Q: How does the decay rate affect the population size in the short term?

A: The decay rate determines the rate at which the population size decreases over time. In the short term, the population size will decrease by a factor of $\sqrt[4]{\frac{1}{2}}$ every hour.

Conclusion

In conclusion, we have answered some frequently asked questions related to the topic of a bacteria population that decreases by a factor of $\frac{1}{2}$ every 4 hours. We hope that this article has provided a better understanding of the concept of population decay and the relationship between the decay rate and the time period.

References

• [1] Population dynamics. Encyclopedia Britannica.
• [2] Exponential decay. Wolfram MathWorld.
• [3] Bacteria population dynamics. Journal of Bacteriology.

Further reading

• [1] Population ecology. Oxford University Press.
• [2] Mathematical modeling of population dynamics. Springer.
• [3] Bacteria population modeling. Journal of Mathematical Biology.