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 \(8.5,8.6)\) COMPUTER PRICES From 1996 to 2000, the average price of a ecomputer company's least expensive home computer system decreased by \(16 \%\) per year.

Short answer

The situation can be represented by an exponential decay model. The model for the situation is \(y = P(1 - 0.16)^t\), with \(y\) being the average price of a computer, \(P\) the initial price, and \(t\) the number of years since 1996.

Step-by-step solution

Identifying the Type of Exponential Model

Since the average price of a computer system is decreasing by a certain percentage each year, this situation represents an exponential decay.

Defining the Initial Value

In order to construct the exponential decay function, one needs to know the initial value of the entity in question which is decreasing. In this problem, it is the initial price of the computer in 1996. Let's denote the initial price (in dollars) as \(P\).

Formulating the Exponential Decay Model

An exponential decay function can be represented as \(y = a(1 - r)^t\), where \(a\) is the initial value, \(r\) is the rate of decrease, and \(t\) is the time. Assuming the average price of a computer decreases by 16% per year from its initial price \(P\), the model is \(y = P(1 - 0.16)^t\), where \(t\) is the number of years since 1996.

Key concepts

Exponential Growth

When learning about the concepts surrounding change over time, it's important to understand the notion of exponential growth as a foundational element. This phenomenon occurs when a quantity increases by a constant percentage every set period of time. The essence of exponential growth is that the growth rate – the speed at which a quantity increases – is proportional to its current size. In other words, the larger the quantity gets, the faster it grows.

A classic example of exponential growth is compound interest in a bank account, where the money grows due to interest accumulating not only on the initial deposit but also on the accumulated interest from previous periods. In algebraic terms, exponential growth can be described by the function \( y = a(1 + r)^t \) where \(a\) represents the initial amount, \(r\) is the growth rate as a decimal, and \(t\) is the time passed.

It's paramount to not confuse this with linear growth, where the rate of increase is constant regardless of the quantity's size – a straight line on a graph rather than the curve characteristic of exponential functions.

Exponential Decay Model

In the world of exponential changes, there is a counterpart to growth known as exponential decay. This process describes a situation in which a quantity decreases by a consistent percentage over equal time intervals. We observe exponential decay in various real-world phenomena, including the depreciation of electronics like computers and smartphones, the cooling of hot substances, and the disintegration of radioactive substances.

An exponential decay model is mathematically depicted by the formula \( y = a(1 - r)^t \), where \(y\) is the final amount after time \(t\), \(a\) is the original amount (often referred to as the 'initial value'), and \(r\) is the decay rate. For instance, in the scenario of the computer prices decreasing by 16% annually, the decay rate (\(r\)) is 0.16, and the model reflects the faster decrease of value the more time has elapsed. It's essential to understand this concept as it differs from simply subtracting a constant value over time, and instead implies a quantity that's diminishing increasingly slowly, creating a curve asymptotically approaching zero on a graph.

Initial Value

The initial value is a term that signifies the beginning amount or size of the quantity being considered before any growth or decay has occurred. The understanding of initial value is critical because it sets the baseline from which change is measured. In exponential functions, whether it is growth or decay, the initial value is denoted by the variable \(a\).

This initial value serves as the starting point on a graph of an exponential function and is the \(y\)-intercept. When modeling real-life situations — like the depreciation of a computer's price — the initial value would be the original price of the computer system. This is the figure from which the rate of decrease (16% per year in our example) is applied. So, if the initial price of the computer was $1,000, the model would start at this point and decrease by 16% of the remaining value each year. To fully understand an exponential scenario, one must first identify this pivotal figure.