Problem 34 - Algebra A Combined Function

Question

Solve. The population of Saint Barthelemy is decreasing according to the formula \(y=y_{0} e^{-0.0034 t} .\) 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 7448 , use the formula to predict the population of Saint Barthelemy in the year 2025 . Round to the nearest whole number. (Source: The World Almanac)

Step-by-Step Solution

Answer

The population will be approximately 7049.

Step 1: Identify Known Values

We begin by identifying the given values from the problem. We know that \( y_0 = 7448 \) (the initial population in 2009) and \( t = 2025 - 2009 = 16 \) years.

Step 2: Substitute Known Values into the Formula

Substitute \( y_0 = 7448 \) and \( t = 16 \) into the formula \( y = y_0 e^{-0.0034 t} \). This gives us the equation: \( y = 7448 \times e^{-0.0034 \times 16} \).

Step 3: Calculate the Exponent

Calculate the exponent \(-0.0034 \times 16 = -0.0544\). Now the equation becomes \( y = 7448 \times e^{-0.0544} \).

Step 4: Find the Value of the Exponential Expression

Calculate \( e^{-0.0544} \) using a calculator. This gives approximately \( 0.9471 \). Substitute back into the equation to get \( y = 7448 \times 0.9471 \).

Step 5: Compute the Final Population

Multiply \( 7448 \) by \( 0.9471 \) to find the population in 2025: \( y \approx 7049.3608 \).

Step 6: Round to the Nearest Whole Number

Round \( 7049.3608 \) to the nearest whole number to find the population of Saint Barthelemy in 2025. The rounded value is 7049.

Key Concepts

Exponential Decay

Exponential decay is a mathematical concept that describes the process of a quantity decreasing at a rate proportional to its current value. It is commonly represented by the formula:
\[ y = y_0 e^{-kt} \] Here:

• onstants \(k\) and \(t\) represent the decay rate and time, respectively.
• The negative exponent \(-kt\) indicates a decrease over time.

This formula is useful in various real-world scenarios, such as calculating population decrease, radioactive decay, and more. By understanding exponential decay, one can predict how a population or other quantity will decrease over a certain period of time.

Initial Population

In the context of a population decay problem, the initial population is the base amount from which decay begins. It's symbolized by \( y_0 \) in the exponential decay formula.
In our example, the initial population of Saint Barthelemy in 2009 is given as 7448.
"Initial" simply means that this is the starting point, or the population size at time zero (year 2009 in this case). Knowing this number is crucial because all future predictions depend on it.
Accurate data regarding the initial population is essential for reliable calculations, as small errors can significantly affect the prediction.

Calculating Future Population

To calculate the future population using the exponential decay formula, follow these steps:

• Identify the known values: Start by noting down your initial population \( y_0 \) and the time span \( t \).
• Substitute the known values: Insert the initial population and time into the formula. In our example, it's like this:\[ y = 7448 \, e^{-0.0034 \times 16} \].
• Calculate the exponent, which is \(-0.0034 \times 16 = -0.0544\).
• Find the exponential expression's value: Compute \( e^{-0.0544} \), approximately 0.9471.
• Final Calculation: Multiply the initial population by this result: \( 7448 \times 0.9471 \approx 7049.3608 \).

Following these steps ensures you have an accurate population forecast for the given year.

Rounding Numbers

Rounding numbers is a useful mathematical technique that simplifies figures, making them easier to understand or communicate.
When solving our problem using the exponential decay formula, we ended up with a population prediction of approximately 7049.3608.
Rounding to the nearest whole number involves examining the decimal component:

• If the first digit after the decimal is 5 or more: Increase the last whole number by 1.
• If it is less than 5: Just drop the decimal part.

Therefore, 7049.3608 rounds to 7049.
This process of rounding ensures that the results are presented in a manageable, comprehensible way.