Problem 10 - Algebra 1: Concepts and Skills

Question

Classify the model as exponential growth or exponential decay. $$ y=55(3)^{t} $$

Step-by-Step Solution

Answer

The given model \(y = 55(3)^t\) is classified as exponential growth.

Step 1: Identifying the parameters

In the provided formula \(y = 55(3)^t\), the initial amount \(a\) is 55 and the rate \(r\) is 3. This is not in the form of either of the growth or decay formulas. Therefore, the rate \(r\) needs to be reformulated.

Step 2: Reformulating the rate

The general models of exponential growth and decay add or subtract the rate \(r\) from 1 respectively. However, in the given model \(y = 55(3)^t\), 3 is the multiplying factor, implying an increase from 1. Thus, the model resembles \(y = 55(1 + r)^t\), where \(r = 3 - 1 = 2\). We are now dealing with a rate of 2.

Step 3: Classifying the model

Having a rate \(r > 0\) suggests that for each time increment, the quantity is multiplied by a factor greater than 1, which describes an increase or growth. Therefore, the model is classified as exponential growth.

Key Concepts

Exponential Decay

Exponential decay is a concept where a quantity decreases over time at a rate proportional to its current value. It's often represented by the formula
\[ y = a(1 - r)^t \]
where

• \( y \) is the final amount,
• \( a \) is the initial amount,
• \( r \) is the decay rate (a number between 0 and 1), and
• \( t \) is time.

In contrast to exponential growth, here the rate \( r \) is subtracted from 1. This signifies a decrease, as the quantity is multiplied by a factor less than 1 as time progresses. Real-world examples of exponential decay include radioactive decay and depreciation of assets over time. Understanding this concept is crucial as it allows students to recognize patterns of decrease in various scientific and financial contexts.

Exponential Functions

Exponential functions are mathematical expressions characterized by a constant rate of growth or decay relative to the function itself. These functions are pivotal in representing real-world situations where change occurs rapidly and non-linearly. The general form of an exponential function is
\[ y = ab^t \]
where

• \( y \) represents the value of the function at time \( t \),
• \( a \) is the initial value,
• \( b \) is the base which determines the direction and rate of growth (if \( b > 1 \)) or decay (if \( 0 < b < 1 \)), and
• \( t \) indicates time or growth periods.

The base of the exponential function differentiates growth from decay. In the textbook's exercise, where the function is \( y = 55(3)^t \), the base is 3 (>1), which indicates growth. Exponential functions can be used to model population growth, investment growth, or even the spread of viruses.

Mathematical Models

Mathematical models serve as simplified representations of real-world phenomena. They use mathematical language to describe patterns, make predictions, and provide insights into complex problems. Exponential functions play a significant role in mathematical modeling, as they capture processes involving swift changes. The process of building a mathematical model includes formulating the model's structure, which is dependent on the observed behavior of the real-world phenomenon.
For instance, the exponential model \( y = 55(3)^t \) in the exercise encapsulates a scenario of continuous growth – this could be related to financial investments, bacteria populations, or other scenarios with increasing patterns. By adjusting the parameters in the exponential function – such as the initial amount \( a \), and the growth rate \( b \) – models can be fine-tuned to replicate observed data accurately and make reliable forecasts. Understanding how to interpret and manipulate these models is a powerful skill, allowing students to analyze and approach problems in a structured manner.