Question

A certain car was purchased for $22,500. The car
has a resale value of $14,500 after a useful life of 6
years. What is the annual depreciation expense for
this car?
Annual depreciation expense = $ [?]
Round to the nearest hundredth.

Answer 1

$\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\dotfill &\ 14500\\ P=\textit{initial amount}\dotfill &\ 22500\\ r=rate\to r\%\to \frac{r}{100}\\ t=years\dotfill &6\\ \end{cases}$

$14500=22500(1 - \frac{r}{100})^{6}\implies \cfrac{14500}{22500}=\left( \cfrac{100-r}{100} \right)^6\implies \cfrac{29}{45}=\left( \cfrac{100-r}{100} \right)^6 \\\\\\ \sqrt[6]{\cfrac{29}{45}}=\cfrac{100-r}{100}\implies 100\sqrt[6]{\cfrac{29}{45}}=100-r \\\\\\ r=100-100\sqrt[6]{\cfrac{29}{45}}\implies r\approx \stackrel{\%}{7.06}$