Question

The number of bank failures decayed approximately exponentially from 157 bank failures in 2010 to 18 bank failures in 2014. Estimate the number of bank failures in2018.

Answer 1

This problem deals with an exponential decay process. The general equation describing a decay is written as

$y=y_0e^{-kt}$

where y0 represents the initial state of the system and k is the decay constant.

In the given problem, we have y0 = 157. After t =4, the value of y becomes 18. We will be using these values to solve first the decay constant of the problem. We have

$\begin{gathered} \frac{y}{y_0}=e^{-kt} \\ -kt=\ln (\frac{y}{y_0}) \\ -kt=\ln (\frac{18}{157}) \\ -kt=-2.16587 \\ k=\frac{-2.16587}{-4}=0.5415 \end{gathered}$

We use this decay constant to solve the number of bank failures in 2018. In the year 2018, t = 8. Hence, the number of bank failures at this time will be

$\begin{gathered} y=(157)e^{-(0.5415\times8)} \\ y=2.063 \\ y\approx2 \end{gathered}$

There will be 2 bank failures in 2018.