Question

How long will it be (hours) until a silo filled with 50 tons of grain is half empty if the silo is emptying at the rate of 4% per hour

Answer 1

`\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\dotfill &\stackrel{half~empty}{25~\hfill }\\ P=\textit{initial amount}\dotfill &50\\ r=rate\to 4\%\to (4)/(100)\dotfill &0.04\\ t=\textit{elapsed time}\\ \end{cases}`

`25=50(1-0.04)^t\implies \cfrac{25}{50}=(1-0.04)^t\implies \cfrac{1}{2}=0.96^t \\\\\\ \log\left( \cfrac{1}{2} \right)=\log(0.96^t)\implies \log\left( \cfrac{1}{2} \right)=t\log(0.96) \\\\\\ \cfrac{\log\left( (1)/(2) \right)}{\log(0.96)}=t\implies 16.9797\approx t\implies \stackrel{hr}{17}\approx t`