Problem 6 - Essential Calculus Early Transcendentals

Question

The table gives the population of India, in millions, for the second half of the 20 th century. $$\begin{array}{|c|c|}\hline \text { Year } & {\text { Population }} \\\ \hline 1951 & {361} \\ \hline 1961 & {439} \\ \hline 1971 & {548} \\ \hline 1981 & {683} \\ \hline 1901 & {846} \\ \hline 2001 & {1029} \\\ \hline\end{array}$$ (a) Use the exponential model and the census figures for 1951 and 1961 to predict the population in \(2001 .\) Com- pare with the actual figure. (b) Use the exponential model and the census figures for 1961 and 1981 to predict the population in \(2001 .\) Com- pare with the actual population. Then use this model to predict the population in the years 2010 and 2020 . (c) Graph both of the exponential functions in parts (a) and (b) together with a plot of the actual population. Are these models reasonable ones?

Step-by-Step Solution

Answer

Complete exponential models and compare predictions to actual data. Analyze model accuracy through comparison and graphing.

Step 1: Understanding Exponential Growth

The exponential growth model can be given by the equation \( P(t) = P_0 e^{kt} \), where \( P_0 \) is the initial population, \( k \) is the growth rate, and \( t \) is the time in years from \( P_0 \). We need to calculate \( k \) using the given data points for each scenario.

Step 2: Calculate Growth Rate for Part (a)

Using the data from 1951 and 1961, we have \( P_0 = 361 \) million at \( t = 0 \) and \( P(t) = 439 \) million at \( t = 10 \) years. Using \( 439 = 361e^{10k} \), solve for \( k \). \[ e^{10k} = \frac{439}{361} \rightarrow 10k = \ln\left(\frac{439}{361}\right) \rightarrow k = \frac{\ln\left(\frac{439}{361}\right)}{10} \].

Step 3: Predict Population for 2001 using Part (a)

Use the value of \( k \) obtained in Step 2 to predict the population for \( t = 50 \) (year 2001). Substitute \( t \) into \( P(t) = 361e^{50k} \) and compute \( P(50) \).

Step 4: Compare Prediction with Actual Population for Part (a)

The model predicts a certain population for 2001. Compare this predicted value with the actual population of 1029 million.

Step 5: Calculate Growth Rate for Part (b)

Using the data from 1961 and 1981, with \( P_0 = 439 \) million at \( t = 0 \) and \( P(t) = 683 \) million at \( t = 20 \), solve \( 683 = 439e^{20k} \) for \( k \). \[ e^{20k} = \frac{683}{439} \rightarrow 20k = \ln\left(\frac{683}{439}\right) \rightarrow k = \frac{\ln\left(\frac{683}{439}\right)}{20} \].

Step 6: Predict Population for 2001 using Part (b)

Use the calculated \( k \) from Step 5 to predict the population in 2001, using \( P(t) = 439e^{40k} \), where \( t = 40 \).

Step 7: Compare Prediction with Actual Population for Part (b)

Compare the predicted population from the model in Step 6 with the actual population of 1029 million.

Step 8: Predict Population for 2010 and 2020 using Part (b)

Using the growth rate from Step 5, predict the population for 2010 (\( t = 49 \)) and 2020 (\( t = 59 \)) using the formula \( P(t) = 439e^{tk} \).

Step 9: Graph the Models and Actual Data

Graph both exponential functions from Parts (a) and (b) along with the actual population data. Analyze if the models match the actual data closely to determine their reasonability.

Key Concepts

Population Prediction

Predicting population growth is an important application of the exponential growth model, especially when it comes to understanding how populations increase over time. This model uses mathematical principles to help forecast future population levels based on past data.

In the case of India, population data from as far back as 1951 helps estimate future numbers. By fitting an exponential function to historic census data, predictions for future years, such as 2001, 2010, and 2020, can be made. The exponential growth formula used is \( P(t) = P_0 e^{kt} \), where,

• \( P_0 \) represents the initial population size.
• \( k \) is the growth rate, a crucial component calculated using given conditions.
• \( t \) is the time in years since the initial measurement.

This prediction process involves determining the rate \( k \) using data like the difference in populations between 1951 and 1961, which is then applied to calculate future values.

Growth Rate Calculation

The growth rate \( k \) is a fundamental element when using the exponential growth model. It gives us a way to understand how quickly a population is growing over time. Calculating \( k \) involves comparing data from two time points.

For example, using population figures from 1951 and 1961, we set up an equation like \( 439 = 361e^{10k} \), representing population increase over 10 years. Solving for \( k \) involves:

• Dividing the later population size by the initial size, resulting in a factor of growth over the period.
• Using logarithms to solve \( e^{10k} = \frac{439}{361} \).
• Finding \( k = \frac{\ln(\frac{439}{361})}{10} \), thereby providing a per year growth rate.

This growth rate is then used to make predictions for future populations.

Exponential Functions Graphing

Graphing exponential functions is a great way to visualize how predicted populations compare with actual census data over time. By plotting functions derived from different time frames, we can assess their accuracy and reasonability.

In this scenario, each exponential model derived from segments of data—such as 1951 to 1961 or 1961 to 1981—is graphed alongside actual data through 2001. Doing so requires plotting both the computed exponential curve and actual population points. We use:

• Different colors or styles to distinguish between historical predictions and real data points.
• Interpreting graphs to understand trends better.

Analyzing these graphs helps determine how closely a model matches reality, revealing its potential accuracy for long-term prediction.