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 \(8.5,8.6)\) Music SALES From 1995 to \(1999,\) the number of CDs a band sold increased by \(23 \%\) per year.

Short answer

The situation can be represented by a model of exponential growth. The model that represents the situation is \(P = P_0(1+0.23)^n\). This implies that the number of CDs a band sells is increasing by \(23%\) each year, starting from an initial number of CDs sold in 1995, \(P_0\).

Step-by-step solution

Identify the Initial Value and Growth Rate

Here, we know that the number of CDs sold each year by a band increases by \(23%\). Unfortunately, we aren't given the initial quantity (the quantity in 1995). This isn't too much of an issue - we can call the initial quantity \(P_0\), and the quantity each next year will still be an increase of \(23%\). So, the growth rate, \(r\), is \(0.23\) (expressing \(23%\) as a decimal).

Apply the Exponential Growth Formula

We apply the exponential growth formula, which is \(P = P_0(1+r)^n\). With \(r = 0.23\) and \(n\) as the number of years since 1995. To solve for any given year after 1995, substitute the value of \(n\) (the number of years since 1995) into the formula.

Write the Exponential Growth Model

Substituting the above identified initial value \(P_0\) and the growth rate into the exponential growth model gives us: \(P = P_0(1+0.23)^n\). This equation allows us to predict the number of CDs the band will sell in any year after 1995, if we know how many they sold in 1995 (which we denote as \(P_0\)).