Question

A property's value is $400,000 and its land value is $75,000. Assuming a depreciation term of 39 years, what is the amount of annual depreciation?

Answer 1

`\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\dotfill &75000\\ P=\textit{initial amount}\dotfill &400000\\ r=rate\to r\%\to (r)/(100)\\ t=years\dotfill &39\\ \end{cases} \\\\\\ 75000=400000(1 - (r)/(100))^(39)\implies \cfrac{75000}{400000}=\left( \cfrac{100-r}{100} \right)^(39)`

`\cfrac{3}{16}=\left( \cfrac{100-r}{100} \right)^(39)\implies \sqrt[39]{\cfrac{3}{16}}=\cfrac{100-r}{100}\implies 100\sqrt[39]{\cfrac{3}{16}}=100-r \\\\\\ r=100-100\sqrt[39]{\cfrac{3}{16}}\implies r\approx 4.2`