Question

EXPONENTIAL MODELS Tell whether the situation can be represented by a model of exponential growth or exponential decay. Then write a model that represents the situation. (Review \(8.5,8.6)\) P. Cooking CLUB From 1996 to 2000, the number of members in the cooking club decreased by \(3 \%\) per year.

Short answer

The situation represents exponential decay and can be modeled by the equation \(N = N_0(1 - 0.03)^t\), where \(N\) is the number of members in a given year, \(N_0\) is the number of members in 1996 and \(t\) is the number of years since 1996.

Step-by-step solution

Identify the Type of Growth

The first step is to identify whether the situation represents exponential growth or decay. Since the number of members in the cooking club has decreased by \(3 \%\) per year over a certain period of time, it is a case of exponential decay.

Construct the Exponential Decay Model

Exponential decay can generally be represented by the formula \(N = N_0(1 - \%\Delta)^t\), where \(N_0\) is the initial amount, \(\%\Delta\) is the percentage change expressed as a decimal, and \(t\) is time. In this case, the percentage change is \(3 \%\) per year, which is expressed as \(0.03\) in decimal form. Since the exact initial number of members is not given, let's represent it as \(N_0\) (members in 1996), thus the formula becomes: \(N = N_0(1 - 0.03)^t\).

Key concepts

Exponential Growth

Exponential growth occurs when a quantity increases by a fixed percentage over equal intervals of time. This can be related to various real-world scenarios like population growth, compound interest in finance, or the spread of a virus. In mathematics, this growth is represented by the equation: \( N = N_0 \times (1 + r)^t \), where \( N \) is the final amount, \( N_0 \) is the initial amount, \( r \) is the growth rate expressed as a decimal, and \( t \) is the time period over which the growth is observed.

For example, if a population of 1,000 bacteria doubles every day, then after 5 days, using the formula with a growth rate of 100% or 1 as a decimal, the population would be \( N = 1000 \times (1 + 1)^5 \). Understanding exponential growth is crucial as it helps predict the future size of growing entities and the resources they might require.

Constructing Exponential Models

Constructing an exponential model requires understanding the nature of the trend you're examining and then encapsulating it into a mathematical framework. An exponential model will take one of two forms, depending on whether it is growth or decay: \( N = N_0 \times (1 + r)^t \) for growth, or \( N = N_0 \times (1 - r)^t \) for decay, where \( N \) is the final amount, \( N_0 \) is the initial amount, \( r \) is the rate expressed as a decimal, and \( t \) is the time.

When constructing these models, converting the growth or decay rate from a percentage to a decimal is essential. This step is commonly overlooked but vital in achieving an accurate model. Additionally, it is imperative to decide on the appropriate time intervals, as these models assume that growth or decay happens continuously at the specified rate.

Percentage Change in Exponential Models

In exponential models, percentage change is the rate at which the quantity grows or decays per time interval. Expressing this rate as a decimal is necessary when plugging it into the exponential equation. In the context of decay, for example, a \( 3\% \) yearly decrease would be written as \( 0.03 \) in decimal form. The model then becomes \( N = N_0 \times (1 - 0.03)^t \) for decay. Understanding percentage change allows for comparing the rate of growth or decay across different scenarios or time frames.

It is equally important to recognize that in exponential decay, even a small percentage change can have a significant effect over time. This attribute of exponential models helps in predicting long-term outcomes, such as the depreciation of an asset or the decrease in the population of a species.