Question

The time, t, required to empty a tank varies inversely as the rate, r, of pumping. If a pump can empty a tank in 2.5 hours at a rate of 400 gallons per minute, how long will it take to empty a tank at 500 gallons per minute?

Answer 1

$\bf \qquad \qquad \textit{inverse proportional variation} \\\\ \textit{\underline{y} varies inversely with \underline{x}} ~\hspace{6em} \stackrel{\textit{constant of variation}}{y=\cfrac{\stackrel{\downarrow }{k}}{x}~\hfill } \\\\[-0.35em] \rule{34em}{0.25pt}$

$\bf \stackrel{\textit{\underline{t} varies inversely with \underline{r}}}{t = \cfrac{k}{r}} \qquad \qquad \qquad \textit{we also know that } \begin{cases} t=2.5\\ r=400 \end{cases} \\\\\\ 2.5=\cfrac{k}{400}\implies 1000=k~\hfill \boxed{t = \cfrac{1000}{r}} \\\\\\ \textit{when r = 500, what is \underline{t}?}\qquad \qquad t = \cfrac{1000}{500}\implies t=2$