Problem 29 - Algebra 1: Concepts and Skills

Question

In Exercises 29 and 30, use the following information. At the start of a basketball tournament consisting of six rounds, there are 64 teams. After each round, one half of the remaining teams are eliminated. Write an exponential decay model showing the number of teams left in the tournament after each round.

Step-by-Step Solution

Answer

The exponential decay model that shows the number of teams left in the tournament after each round is \(N = 64 \cdot (0.5)^t\), where \(t\) is the round number.

Step 1: Identify the Base of the Exponential Decay

The number of teams is halved after each round, so the base of the exponential decay is \(0.5\).

Step 2: Identify the Initial Value

Initially, there are 64 teams. This is the initial amount, denoted as \(N_0\). So, \(N_0 = 64\).

Step 3: Formulate Exponential Decay Function

The general form of an exponential decay function is \(N = N_0 \cdot b^t\), where \(N_0\) is the initial quantity, \(b\) is the decay factor (base), and \(t\) is the number of time periods. Substituting \(N_0 = 64\) and \(b = 0.5\), we get \(N = 64 \cdot (0.5)^t\).

Step 4: Simplification of the Exponential Decay Function

In the case, where \(t\) is the round number, the function is simplified to \(N = 64 \cdot (0.5)^t\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions used to describe processes that grow or decay at a constant percentage rate per time period. They are characterized by the presence of an exponent, typically written in the form:

• Growth form: \( N = N_0 \cdot (1 + r)^t \)
• Decay form: \( N = N_0 \cdot (1 - r)^t \)

In both forms, \( N_0 \) represents the initial amount, \( r \) is the rate of growth or decay, and \( t \) symbolizes the time period.
Exponential decay is a specific type of exponential function where the quantity reduces over time. A common example is the halving of a substance with a known rate of decay, such as radioactive isotopes or, in this case, a sports tournament where half the teams are eliminated each round.
Understanding how to manipulate and apply exponential functions is crucial in various fields such as finance, population studies, and physics, where change is not linear but exponential.

Decay Model

A decay model is a mathematical representation used to describe processes that involve a reduction over time, operating at a constant proportional rate.
For instance, in the context of our exercise, the decay model takes the form of an exponential decay equation, expressed as \( N = N_0 \cdot (b)^t \). Here, \( N_0 \) is the initial quantity (in our case, 64 teams), \( b \) is the decay factor, and \( t \) is the time variable quantified as the number of periods or rounds.

• The decay factor \( b \) is crucial. For the given exercise, the factor is 0.5 because each round eliminates half the teams.
• Every time period (each round), the quantity of interest (number of teams) is multiplied by 0.5, showcasing an exponential decay.

The decay model is instrumental in predicting and understanding scenarios where things reduce by a consistent percentage, such as understanding how quickly a tournament narrows down the competitors, or how fast a population or substance decreases.

Mathematical Modeling

Mathematical modeling involves the translation of real-world problems into mathematical expressions and formulas, thereby enabling the prediction and analysis of outcomes.
In our basketball tournament scenario, mathematical modeling helps us analyze how teams are reduced after each round using an exponential decay function. This model allows us to predict how many teams will remain at any point in the tournament:

• We start with a clear understanding of the rules: half of the teams are eliminated each round.
• From this, we create an equation that reflects this behavior: \( N = 64 \cdot (0.5)^t \).

Mathematical models are valuable tools used beyond just tournaments. They are crucial in economics for predicting market trends, in engineering for system designs, and in biology for population studies, among many others. They simplify complex problems, making them accessible for analysis and decision-making. Understanding and applying these models is a key skill in solving real-world problems.