Depreciation of a car

Text

A car is worth $24\,000$ € and each year loses $15\%$ of the value it had the previous year.

• (a) How much is it worth after $5$ years?
• (b) After how many years does the value drop below $10\,000$ €?

Solution

Each year the value is multiplied by $0.85$: it is a geometric progression $V_n=24\,000\cdot 0.85^{\,n}$. (a) $V_5=24\,000\cdot 0.85^{5}\approx 24\,000\cdot 0.4437\approx 10\,648.93\ \text{€}.$ (b) Impose $24\,000\cdot 0.85^{\,n}<10\,000$, i.e. $0.85^{\,n}<0.4167$; taking logarithms, $n>\frac{\ln 0.4167}{\ln 0.85}\approx 5.39,$ so the first whole year that works is $n=6$ (then $V_6\approx 9052\ \text{€}$). $\boxed{\,V_5\approx 10\,648.93\ \text{€},\quad n=6\ \text{years}\,}$