Question

Alexa purchased a new car valued at $30,000 that depreciated continuously at a rate

of 25%. Its current value is $12,000. The equation 12000 = 30000(1 r)t
represents the situation, where t is the age of the car in years and r is the rate of
depreciation. About how old is Alexa's car? Round your answer to the nearest year.
• 2 years
• No Solution
O 1 year
• 3 years
O 5 years

Answer 1

`\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\dotfill &\$12000\\ P=\textit{initial amount}\dotfill &\$30000\\ r=rate\to 25\%\to (25)/(100)\dotfill &0.25\\ t=years\\ \end{cases}`

`12000=30000(1 - 0.25)^(t)\implies \cfrac{12000}{30000}=(1 - 0.25)^(t)\implies \cfrac{2}{5}=0.75^t \\\\\\ \log\left( \cfrac{2}{5} \right)=\log(0.75^t)\implies \log\left( \cfrac{2}{5} \right)=t\log(0.75) \\\\\\ \cfrac{~~\log\left( (2)/(5) \right)~~}{\log(0.75)}=t\implies 3\approx t`