Question

Finding the time to reach a limit in a word problem on...

A laptop computer is purchased for $3300. Each year, its value is 75% of its value the year before. After how many years will the laptop computer be worth
$500 or less? (Use the calculator provided if necessary.)
Write the smallest possible whole number answer.
years

Answer 1

`\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\dotfill & \$ 500\\ P=\textit{initial amount}\dotfill &3300\\ r=rate\to 75\%\to (75)/(100)\dotfill &0.75\\ t=years \end{cases}`

`500 = 3300(1 - 0.75)^(t) \implies \cfrac{500}{3300}=0.25^t\implies \cfrac{5}{33}=0.25^t \\\\\\ \log\left( \cfrac{5}{33} \right)=\log(0.25^t)\implies \log\left( \cfrac{5}{33} \right)=t\log(0.25) \\\\\\ \cfrac{\log\left( (5)/(33) \right)}{\log(0.25)}=t\implies 2\approx t\qquad \qquad \textit{to be exact, about 1 year and 131 days}`