Question

A business earned \(\$ 85,000\) in 1990 . Then its earnings decreased by \(2 \%\) each year for 10 years. Write an exponential decay model for the earnings \(E\) in year \(t\). Let \(t=0\) represent 1990 .

Short answer

The exponential decay model for the earnings is \(E = 85000 (1 - 0.02)^t\) where \(t\) stands for years after 1990.

Step-by-step solution

Identify the Initial Value

The initial value or the principal \(P\) in this case is given as the earnings of the business in the year 1990, which is \$85,000.

Identify the Rate of Decay

The percentage decrease each year, which forms our rate of decay \(r\), is stated in the question as \(2\%\). We convert this percentage to a decimal by dividing by 100. Thus, \(r = 2/100 = 0.02\).

Formulate the Exponential Decay Model

Substitute the value of \(P\) and \(r\) into the general form of the exponential decay equation \(E = P (1 - r)^t\). Upon substitution, we get the model: \(E = 85000 (1 - 0.02)^t\). Note that \(t = 0\) corresponds to the year 1990, and \(t = 10\) corresponds to the year 2000.