Chapter 4.2, Problem 32PS

Solution

To find: Show the ratio of depreciation amounts for consecutive years is constant then write an equation that gives $d$ as a function of $t$

The equation becomes $d=1985{\left(0.96\right)}^t$

Given information:

Given that

Depreciation amount for consecutive years is below:

Year (t)	1	2	3	5	5
Depreciation(d)	1906	1832	1762	1692	1627

Calculation:

The exponential decay model is $d=a{\left(1-r\right)}^t$ because there is depreciation.

Now find the ratio of depreciation between consecutive years:

  $\begin{array}{l}\frac{year2}{year1}=\frac{1832}{1906}\approx 0.96\\ \frac{year3}{year2}=\frac{1762}{1832}\approx 0.96\\ \frac{year4}{year3}=\frac{1692}{1762}\approx 0.96\\ \frac{year5}{year4}=\frac{1627}{1692}\approx 0.96\end{array}$

Therefore the decay factor $1-r=0.96$

Now the initial amount when $t=0$ will be:

  $\begin{array}{l}1906=a\left(0.96\right)\\ a=\frac{1906}{0.96}\\ a\approx 1985\end{array}$

In the decay model $d=a{\left(1-r\right)}^t$

  $d$ is the depreciation amount, $a$ is initial amount and $1-r$ is decay factor which is $0.96$ .The equation becomes:

  $d=1985{\left(0.96\right)}^t$ .

This is the required equation.