Problem 51 - Calculus

Question

In April 2015 in the US, there was one birth every 8 seconds, one death every 12 seconds, and one new international migrant every 32 seconds. 13(a) Let \(f(t)\) be the population of the US, where \(t\) is time in seconds measured from the start of April \(2015 .\) Find \(f^{\prime}(0) .\) Give units. (b) To the nearest second, how long did it take for the US population to add one person in April 2015?

Step-by-Step Solution

Answer

(a) \( f'(0) = \frac{7}{96} \text{ people/second} \). (b) It takes about 14 seconds.

Step 1: Determine Birth Rate Contribution

Given that there was one birth every 8 seconds, calculate the rate at which births contribute to the population increase. This rate is given by \( \frac{1 \, \text{birth}}{8 \, \text{seconds}} = \frac{1}{8} \, \text{people/second} \).

Step 2: Determine Death Rate Contribution

Given that there was one death every 12 seconds, calculate the rate at which deaths contribute to the population decrease. This rate is given by \( -\frac{1 \, \text{death}}{12 \, \text{seconds}} = -\frac{1}{12} \, \text{people/second} \).

Step 3: Determine Migration Rate Contribution

Given that there is one new international migrant every 32 seconds, calculate the rate at which migration contributes to the population increase. This rate is \( \frac{1 \, \text{migrant}}{32 \, \text{seconds}} = \frac{1}{32} \, \text{people/second} \).

Step 4: Calculate Total Population Change Rate \( f'(t) \)

The rate of change of the population, \( f'(t) \), is the sum of the rates determined in steps 1, 2, and 3:\[ f'(t) = \frac{1}{8} - \frac{1}{12} + \frac{1}{32} \] Calculate \( f'(0) \). We find the common denominator, which is 96. The expression becomes:\[f'(0) = \frac{12}{96} - \frac{8}{96} + \frac{3}{96} = \frac{7}{96} \text{people/second}\] Thus, \( f'(0) = \frac{7}{96} \) people per second.

Step 5: Calculate the Time to Increase Population by One Person

Find how long it takes for the population to increase by one person. Since the rate of change is \( \frac{7}{96} \) people per second, set up the equation:\[\frac{7}{96} \times t = 1\]Solve for \( t \):\[t = \frac{96}{7} \approx 13.71 \text{ seconds}\] Rounded to the nearest second, it takes approximately 14 seconds to add one person to the population.

Key Concepts

Rate of Change

Population growth is generally analyzed by looking at the rate of change. This involves understanding how quickly the population number is increasing or decreasing over time. The rate of change is calculated by summing different factors influencing population dynamics: birth rates, death rates, and migration rates. For the given exercise, the rate of change is symbolized as \( f'(t) \). The calculated value indicates how many people are added to or subtracted from the population every second.
The formula used in the exercise to find \( f'(0) \), which represents the rate of change at the initial time, is:

• Birth rate contribution: \(\frac{1}{8}\) per second
• Death rate contribution: -\(\frac{1}{12}\) per second
• Migration rate contribution: \(\frac{1}{32}\) per second

Adding these three components gives the overall population change rate per second: \[ f'(0) = \frac{7}{96} \text{ people/second} \].
This means that in every second, approximately \( \frac{7}{96} \) of a person is added to the U.S. population during the calculation period.

Birth Rate

The birth rate is a crucial determinant of population growth. It signifies how many people are born per unit of time, typically measured annually or, like in the exercise, more precisely per second. In the exercise, it was given that one birth occurs every 8 seconds in the United States.
To express this as a rate, we find that:

• 1 birth every 8 seconds is equal to \(\frac{1}{8}\) births per second.

This information was used to determine how births contribute to the population's overall growth rate, helping in summing up the components contributing to the rate of change of the population. This indicates the constant inflow contributing positively to changing population numbers.

Death Rate

The death rate is equally important in the overall calculation of population dynamics, as it subtracts from the population count. In the context of the exercise, it is noted that one death occurred every 12 seconds. This presents a negative contribution to the population as each death reduces the total number of people.
The contribution of deaths is thus calculated as:

• 1 death every 12 seconds corresponds to \(-\frac{1}{12}\) deaths per second.

The negative sign here is crucial; it shows that this factor decreases the population's total size. By integrating the effect of the death rate into the overall rate of change, we can accurately gauge the real-time population dynamics.

Migration

Migration plays a unique role in population changes, representing individuals moving across borders. For the United States, the exercise describes that there is one new international migrant every 32 seconds.
This adds another positive component to the rate of change, similar to births but typically at a different rate:

• 1 migrant every 32 seconds equals \(\frac{1}{32}\) migrants per second.

This rate indicates how migration can sustain or even increase population size by adding more people to the country. Understanding migration is vital, as it can significantly alter demographic compositions and influence socio-economic factors in the long term. Combined with birth and death rates, it helps form a comprehensive view of population changes.