Question

the time it takes to fill a water tank varies inversely with the water rate of the hose. At 8 gallons per minute, a hose can fill the tank in 45 minutes. How long will it take to fill the same tank at 15 gallons per minute

Answer 1

$\bf \qquad \qquad \textit{inverse proportional variation}\\\\ \textit{\underline{y} varies inversely with \underline{x}}\qquad \qquad y=\cfrac{k}{x}\impliedby \begin{array}{llll} k=constant\ of\\ \qquad variation \end{array}\\\\ -------------------------------\\\\ \textit{time it takes to fill a tank varies inversely with the water rate}\\\\ t=\cfrac{k}{w}\quad \begin{cases} t=\textit{time to fill}\\ w=\textit{water rate} \end{cases}$

$\bf \textit{we also know that } \begin{cases} w=\stackrel{g/min}{8}\\ t=\stackrel{min}{45} \end{cases}\implies 45=\cfrac{k}{8}\implies 360=k \\\\\\ therefore\qquad \boxed{t=\cfrac{360}{w}} \\\\\\ \textit{now if w = 15, what is \underline{t}?}\qquad t=\cfrac{360}{15}$