Problem 33 - Algebra A Combined Function

Question

Solve. The population of the Cook Islands is decreasing according to the formula \(y=y_{0} e^{-0.0277 t}\). In this formula, \(t\) is time in years and \(y_{0}\) is the initial population at time 0 . If the size of the population in 2009 was \(11,870,\) use the formula to predict the population of Cook Islands in the year \(2025 .\) Round to the nearest whole number. (Source: The World Almanac)

Step-by-Step Solution

Answer

The estimated population of the Cook Islands in 2025 is 7,612.

Step 1: Identify Known Values

We know the initial population at the starting year (2009) is given as \( y_0 = 11,870 \). The target year is 2025, which means \( t = 2025 - 2009 = 16 \) years. The decay rate from the formula is \( -0.0277 \).

Step 2: Substitute Into Formula

Substitute the known values into the population decay formula: \[ y = 11,870 \times e^{-0.0277 \times 16} \]

Step 3: Calculate Exponential Term

Calculate the exponential component of the formula:\[ e^{-0.0277 \times 16} \approx e^{-0.4432} \] Using a calculator, find \( e^{-0.4432} \approx 0.6415 \).

Step 4: Final Calculation

Substitute back into the formula to find the population:\[ y = 11,870 \times 0.6415 \approx 7,612 \]

Step 5: Round to the Nearest Whole Number

The population predicted for 2025 after rounding is approximately 7,612.

Key Concepts

Population Modeling

Population modeling involves using mathematical formulas or equations to estimate how populations change over time. One common reason to study populations is to understand their growth or decline. This can help governments and organizations plan for future needs, such as healthcare, infrastructure, and resources.
In population modeling, the conditions of the environment and historical data are used to predict future populations. The formula given in our example—\( y = y_{0} e^{-0.0277 t} \)—is an exponential decay model. This formula predicts how the population decreases over time. The formula assumes a continuous decline at a constant rate.
Using historical data, such as the initial population of 11,870 in 2009 for the Cook Islands, and by plugging in the time period \( t \) into the formula, predictions can be made for future years like 2025. This aids in strategic planning and decision-making to manage the anticipated population size.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These types of functions are important in modeling real-world phenomena such as interest in finance, radioactive decay, and population change. In our context, the exponential function represents population decay.
The general form is \( y = y_{0} e^{kt} \), where \( y_{0} \) is the initial amount, \( e \) is the base of natural logarithms (approximately 2.71828), \( k \) is the growth (or decay) rate, and \( t \) is time.
When \( k \) is a negative number, as in the population formula \( y = y_{0} e^{-0.0277 t} \), the function models exponential decay, meaning the quantity is decreasing over time. This property makes it perfect for predicting how something like a population will shrink under consistent decay conditions.

Rounding Numbers

Rounding numbers is a mathematical technique used to simplify complex calculations and to express numbers in a more readable form. It is crucial in population predictions to provide meaningful and realistic insights.
The rounding process involves altering a number to be closer to its nearest whole number or decimal based on specified rules—typically by looking at the digit in the tenths place. In this exercise, once the population number was calculated as approximately 7,612, it's rounded to the nearest whole number to provide clarity and practicality.
Rounding not only simplifies communication of numerical data but also ensures that predictions are usable for planning and decision-making, especially when dealing with large numbers such as population estimates.