Problem 186 - Precalculus

Question

For the following exercises, use this scenarion of a city has fisen over a 20 -year interval. Its population may be modeled by the following function: \(y=12,000+8,000\) sin \((0.628 x),\) where the domain is the years since 1980 and the range is the population of the city. What is the largest and smallest population the city may have?

Step-by-Step Solution

Answer

The largest population is 20,000 and the smallest is 4,000.

Step 1: Understand the Sine Function's Range

The sine function, \( \sin(\theta) \), oscillates between -1 and 1. This is important when analyzing the given function in the population model.

Step 2: Analyze the Population Function

The population function is given by \( y = 12,000 + 8,000 \sin(0.628x) \). Here, \( 12,000 \) is the baseline population, and \( 8,000 \sin(0.628x) \) represents the fluctuating part influenced by the sine wave.

Step 3: Determine the Minimum Population

When \( \sin(0.628x) = -1 \), the function reaches its minimum value. Substitute into the function to get: \( y_{\text{min}} = 12,000 + 8,000(-1) = 4,000 \).

Step 4: Determine the Maximum Population

When \( \sin(0.628x) = 1 \), the function reaches its maximum value. Substitute into the function to get: \( y_{\text{max}} = 12,000 + 8,000(1) = 20,000 \).

Key Concepts

Population Modeling

Population modeling helps us understand how populations change over time, often used to predict future trends. In this scenario, a city’s population over a 20-year period is represented by a mathematical model.
The model uses a sine function to drive these predictions, capturing cyclical patterns in population shifts.

• The base population, an average around which the population fluctuates, is 12,000.
• The sine function introduces variability, simulating the natural ebb and flow of migration or other population factors.

This model allows for an analysis of peak growths and declines by calculating maximum and minimum values within the given population range.

Sine Function

The sine function, noted as \( \sin(\theta) \) in mathematics, is a periodic function with a wave-like pattern that oscillates between -1 and 1. It is used to model recurring cycles.
In population modeling, the sine function serves to simulate cycles that may occur due to seasonal changes, employment opportunities, or urban policies.

• The expression \( \sin(0.628x) \) modifies the standard sine wave cycle to fit a particular timeframe, which in this case, corresponds to yearly changes over a 20-year span.
• Inside the population function \( y = 12,000 + 8,000 \sin(0.628x) \), it indicates the dynamics of population fluctuations about the base population of 12,000.

Overall, the sine function is essential for conveying periodic changes within mathematical models of real-world situations.

Amplitude

Amplitude refers to the height of a wave from its centerline to its peak or trough. In the context of the sine function, which is applied to population modeling here, the amplitude controls the extent of fluctuation around the base value.
The amplitude in the population model is 8,000, corresponding to the coefficient of the sine function: \( 8,000 \sin(0.628x) \).

• This tells us that population figures can increase or decrease by 8,000 from the baseline of 12,000.
• Such an amplitude leads to maximum and minimum population calculations: from a low of 4,000 to a high of 20,000.

The amplitude thus quantifies the magnitude of population variation, bridging the gap between static base population and dynamic shifts as influenced by external factors.