Question

The population of a culture of bacteria has a growth rate given by \(p^{\prime}(t)=\frac{200}{(t+1)^{r}}\) bacteria per hour, for \(t \geq 0,\) where \(r > 1\) is a real number. In Chapter 6 it is shown that the increase in the population over the time interval \([0, t]\) is given by \(\int_{0}^{t} p^{\prime}(s) d s\). (Note that the growth rate decreases in time, reflecting competition for space and food.) a. Using the population model with \(r=2,\) what is the increase in the population over the time interval \(0 \leq t \leq 4 ?\) b. Using the population model with \(r=3,\) what is the increase in the population over the time interval \(0 \leq t \leq 6 ?\) c. Let \(\Delta P\) be the increase in the population over a fixed time interval \([0, T] .\) For fixed \(T,\) does \(\Delta P\) increase or decrease with the parameter \(r ?\) Explain. d. A lab technician measures an increase in the population of 350 bacteria over the 10 -hr period [0,10] . Estimate the value of \(r\) that best fits this data point. e. Looking ahead: Use the population model in part (b) to find the increase in population over the time interval \([0, T],\) for any \(T > 0 .\) If the culture is allowed to grow indefinitely \((T \rightarrow \infty)\) does the bacteria population increase without bound? Or does it approach a finite limit?

Short answer

b) What is the increase in population for t = 0 to t = 6 with r = 3? c) How does the increase in population (\(\Delta P\)) behave with respect to the parameter r? d) What is the estimated value of r that best fits the given data point, given an increase in population of 350 bacteria over a 10-hour period? e) Does the increase in population become unbounded or approach a finite limit as time tends to infinity?

Step-by-step solution

a) Calculate the increase in population for t = 0 to t = 4 with r = 2

Using the given formula for the growth rate - \(p^{\prime}(t)=\frac{200}{(t+1)^{r}}\), we will set r = 2 and integrate for the time interval \([0, 4]\). \(\Delta P = \int_{0}^{4} \frac{200}{(t+1)^{2}} dt\)

b) Calculate the increase in population for t = 0 to t = 6 with r = 3

Similarly, we will set r = 3 and integrate for the time interval \([0, 6]\). \(\Delta P = \int_{0}^{6} \frac{200}{(t+1)^{3}} dt\)

c) Behavior of \(\Delta P\) based on parameter r

To understand the relationship between \(\Delta P\) and r, we will take the derivative of \(\Delta P\) with respect to r. The sign of the derivative will help us determine how \(\Delta P\) changes with r. \(\frac{d \Delta P}{d r} = -\int_{0}^{T} 200 \times ln(t+1)(t+1)^{-r} dt\)

d) Estimate the value of r that best fits the given data point

Given the information that there is an increase in population of 350 bacteria over a 10-hour period, we need to find the value of r for which: \(\Delta P = 350 = \int_{0}^{10} \frac{200}{(t+1)^{r}} dt\)

e) Analyze the increase in population for T > 0 and T \(\rightarrow \infty\)

We will use the population model from part b: \(\Delta P = \int_{0}^{T} \frac{200}{(t+1)^{3}} dt\) We will evaluate the limit as T approaches infinity and determine if the population increase is unbounded or finite: \(\lim_{T\to\infty} \Delta P = \lim_{T\to\infty} \int_{0}^{T} \frac{200}{(t+1)^{3}} dt\)

Key concepts

Differential Equations

Differential equations are a cornerstone in understanding real-world phenomena where one quantity changes in relation with another. In our bacteria population problem, the growth rate, noted as p'(t), represents the derivative of the population with respect to time t. Essentially, a differential equation has variables and their derivatives, and solving these helps us understand how a system evolves over time.

In the context of the exercise, we’re given a differential equation with a variable growth rate that decreases as time progresses. To solve for the increase in the population over a time interval, we apply the concept of indefinite integrals to find the function representing the total population.

Indefinite Integrals

Indefinite integrals are essentially the reverse of derivatives and are used to find antiderivatives. They’re a crucial part of solving differential equations, as they help us determine the original function before differentiation. For example, integrating the bacteria growth rate given in our problem over a specific time interval gives us the net increase in population during that time.

In the exercise, the integral of p'(t) with respect to time t from 0 to T is calculated to find the total change in population, ΔP. This is done by computing the area under the curve of the growth rate function over the specified interval, which is a classic application of indefinite integrals.

Limits in Calculus

Limits in calculus help us understand the behavior of functions as they approach a specific point or extend indefinitely. They’re essential in determining continuity and for evaluating functions whose direct computation is not possible. In our bacteria culture problem, the limit concept is invoked to analyze the long-term behavior of the population growth.

Specifically, to answer part e of the exercise, we use limits to determine what happens to the bacteria population as T approaches infinity. The limit of the integral of the growth rate helps us conclude whether the population will continue to increase endlessly (unbounded growth) or approach a finite value (bounded growth), critical for understanding capacity constraints in an ecological context.

Exponential Decay

Exponential decay describes the process where quantities decrease at a rate proportional to their current value. This idea is reflected in our population model, where the growth rate decreases over time, indicating competition for limited resources. Exponential decay is commonly seen in radioactive decay, cooling of substances, and in this case, population dynamics.

In the exercise, we observe the model’s decay behavior, as the growth rate’s denominator has an exponent r that causes the rate to decay as time t increases. Understanding exponential decay allows us to better predict how the population will develop over time, manage resources, and estimate parameters like the decay factor r for different scenarios, such as in part d of the given problem.