Problem 6 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

There is a suggestion that the world human population is growing exponentially. Shown below are the human population numbers in billions of people for the decades \(1940-2010\). \(\begin{array}{lrrrrrrrr}\text { Year } & 1940 & 1950 & 1960 & 1970 & 1980 & 1990 & 2000 & 2010 \\ \text { Index, } t & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \text { Human Population } \times 10^{6} & 2.30 & 2.52 & 3.02 & 3.70 & 4.45 & 5.30 & 6.06 & 6.80\end{array}\) 1\. Test the equation $$ P_{t}=2.21 .19^{t} $$ against the data where \(t\) is the time index in decades after 1940 and \(P_{t}\) is the human population in billions. 2\. What percentage increase in human population each decade does the model for the equation assume? 3\. What world human population does the equation predict for the year \(2050 ?\)

Step-by-Step Solution

Answer

1. The model slightly overestimates real data; 2. A 19% increase each decade; 3. Predicted 2050 population: 12.55 billion.

Step 1: Evaluate the Equation for Each Year

For each decade, calculate the expected population using the equation \( P_{t} = 2.21 \times 1.19^{t} \). Perform this calculation for all \( t \) from 0 to 7 (1940 to 2010) and compare it to the actual given data. For instance, for 1940, \( P_{0} = 2.21 \times 1.19^{0} = 2.21 \) billion.

Step 2: Calculate Expected Values and Compare

Continue the calculations for each index:- 1950: \( P_{1} = 2.21 \times 1.19^{1} = 2.63 \) billion.- 1960: \( P_{2} = 2.21 \times 1.19^{2} = 3.12 \) billion.- 1970: \( P_{3} = 2.21 \times 1.19^{3} = 3.71 \) billion.Compare these against the actual data to check accuracy.

Step 3: Analyze Model's Performance

Check each calculated population value against the actual data: - 1940 was predicted 2.21 vs actual 2.30 billion (close). - 1950 predicted 2.63 vs actual 2.52 billion (close). - 1960 predicted 3.12 vs actual 3.02 billion (close). - Continue this for 1970-2010.

Step 4: Determine Percentage Increase per Decade

The model assumes population grows by a factor of 1.19 each decade, which represents a \(19\%\) increase per decade. This is deduced from the base of the exponential, \(1.19\), representing \(1 + r\) where \(r\) is the growth rate.

Step 5: Predict Future Population for 2050

To estimate the population in 2050, calculate for \( t = 11 \) since 2050 is 11 decades after 1940:\[ P_{11} = 2.21 \times 1.19^{11} \approx 12.55 \text{ billion} \]This is the predicted population for 2050 using the model.

Key Concepts

Population Modeling

Population modeling is a mathematical way to represent how a population changes over time. In this case, we're dealing with the human population of the Earth from 1940 to 2010. Exponential growth is a common model used to describe populations that increase rapidly over time. In such models, the population in the future is predicted based on its current size multiplied by a growth factor associated with each time period.

For our exercise, the population is predicted using the equation \(P_{t} = 2.21 \times 1.19^{t}\). Here, \(P_{t}\) represents the population in billions at time \(t\) decades after 1940. The base value \(2.21\) stands for the population in 1940, while 1.19 is the growth factor indicating a \(19\%\) increase each decade.

This simple equation allows us to track the exponential pattern of human population growth. By adjusting the variable \(t\), we can observe how well the model fits actual historical data and make predictions about the future, effectively enabling us to 'see' the population trend and deviations from it over this 70-year span. Understanding these principles is key to solving and applying such mathematical models in real-world scenarios.

Growth Rate Calculation

The growth rate calculation is essential for understanding how fast a population is expanding over time. The growth rate \(r\), when modeling exponential growth, depicts the percentage increase in population per unit of time—in our exercise, per decade.

The equation used, \(P_{t} = 2.21 \times 1.19^{t}\), includes a base of 1.19. This figure derives from \(1 + r\), where 1 accounts for the current population, and the additional \(0.19\) represents the \(19\%\) growth rate. Consequently,

• The growth rate per decade is \(19\%\), indicating that with each passing decade, the population is expected to be \(19\%\) larger than the previous decade.

This approach provides an intuitive understanding of how small percentage increases can lead to significant population growth over extended periods, reflective of exponential growth behavior. In practical terms, finding \(r\) allows researchers and policymakers to plan for future challenges and opportunities related to population increases.

Predictive Analysis

Predictive analysis involves utilizing mathematical equations to foresee future outcomes based on current and historical data. In our population exercise, this concept is applied to forecast the human population in the year 2050.

By using the equation \(P_{t} = 2.21 \times 1.19^{t}\), we can extend past trends into future decades. For predicting the year 2050, we calculate \(t = 11\) because 2050 is 11 decades from the base year, 1940. Inserting this into the equation gives us:

• \(P_{11} = 2.21 \times 1.19^{11} \approx 12.55 \text{ billion}\).

Predictive analysis like this helps in planning for the implications of such population growth, like resource allocation, urban planning, and environmental protection. This foresight-driven approach is invaluable for anticipating needs and making informed decisions before changes occur.