Question

Use the following information. Each year in the month of March, the NCAA basketball tournament is held to determine the national champion. At the start of the tournament there are 64 teams, and after each round, one half of the remaining teams are eliminated. Write an exponential decay model showing the number of teams \(N\) left in the tournament after round \(t.\)

Short answer

The exponential decay model that shows how many teams are left after round \(t\) is \(N = 64 \times (1/2)^t\).

Step-by-step solution

Understanding the problem

At the start of the tournament there are 64 teams and in each new round, half of the remaining teams are eliminated. So the goal is to find how many teams \(N\) are left after round \(t\). This is a clear scenario of exponential decay.

Formulate the general exponential decay model

The general exponential decay model is of the form \(N = N_0 \times (1 - r)^t\), where \(N_0\) is the initial amount, \(r\) is the decay rate and \(t\) is the time(i.e., the number of rounds in this context).

Apply the problem's specifics to the general model

The beginning number of teams, or \(N_0\), is 64. The rate at which teams are eliminated, or \(r\), is 1/2. This reduction by a half in each round leads to a more simple exponential decay model in this case, \(N = N_0 \times (1/2)^t\). The model becomes: \(N = 64 \times (1/2)^t\).

Key concepts

Exponential Functions

When we talk about exponential functions, we're dealing with a special type of function where a constant base is raised to a variable exponent. This specific action creates a rapid increase or decrease—the 'exponential' part of the name—in the function's output values. An exponential decay function, in particular, describes a situation where the amount decreases by a consistent percentage rate over equal time intervals.

For instance, if a quantity halves over a set time period, it's decreasing exponentially. The formula that represents this decay is typically written as \(N = N_0 \times a^t\), where:\

• \(N\) represents the final amount,
• \(N_0\) is the initial quantity,
• \(a\) is the base of the exponential which is the factor the quantity decreases by each period (less than 1 for decay),
• and \(t\) is the variable representing time or, in different contexts, the number of intervals.

In our NCAA tournament example, the number of teams left after each round of the tournament can be modeled by this type of function.

Algebraic Modeling

Algebraic modeling is the process of representing real-world scenarios with mathematical equations and expressions. It's a powerful technique that transforms observations into soluble mathematical problems. When we construct algebraic models, we use variables, constants, and algebraic symbols to mirror the structure and behavior we observe.

In the case of the NCAA basketball tournament, we can model the predictable halving of teams after each round using variables and constants. The variable \(t\) is crucial as it represents the discrete rounds in the tournament, while the constant \(N_0\) signifies the known starting point, which is 64 teams in this instance. By coupling our understanding of how exponential functions work with the specifics of the tournament structure, we can create an algebraic representation of the decay in the number of teams—a truly practical use of algebra in real life.

Mathematical Representation

The heart of mathematical representation is translating situations, patterns, or relationships into a mathematical language. In this scenario, our goal is to capture the decline in the number of teams remaining in the tournament in a mathematical form. For this, we not only choose the correct type of function—an exponential decay function—but also identify and correctly insert the specific parameters of our real-world example.

The exponential decay model \(N = 64 \times (1/2)^t\) is the mathematical representation we seek. It encapsulates the initial condition of 64 teams, the repeated halving after each round (represented by \((1/2)^t\)), and allows us to plug in any round number \(t\) to determine the number of teams remaining. This mathematical model gives us a succinct and accurate method to predict outcomes, reinforcing the value of mathematics as a predictive and descriptive tool.