Question

A quantity with an initial value of 230 decays continuously at a rate of 7% per year. What is the value of the quantity after 8.7 decades, to the nearest hundredth?

Answer 1

well, there are 10 years in 1 decade, so 8.7 decades will just be 8.7*10 = 87 years, so

`\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\\ P=\textit{initial amount}\dotfill &230\\ r=rate\to 7\%\to (7)/(100)\dotfill &0.07\\ t=years\dotfill &87\\ \end{cases} \\\\\\ A=230(1 - 0.07)^(87)\implies A=230(0.93)^(87)\implies A\approx 0.42`