[tex]\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}[/tex]
[tex]\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[/tex]
Variation can be direct, inverse and joint.
It will take 2 hours to empty the tank at a rate of 500 gallons per minutes
Let time be represented with t, and r represent rate.
The inverse variation is represented as:
[tex]\mathbf{t \alpha \frac{1}{r}}[/tex]
Represent as an equation
[tex]\mathbf{t = \frac{k}{r}}[/tex]
Make k the subject
[tex]\mathbf{k = rt}[/tex]
When t = 2.5 and r = 400, we have:
[tex]\mathbf{k =400 \times 2.5}[/tex]
[tex]\mathbf{k =1000}[/tex]
When r = 500, we have:
[tex]\mathbf{k = rt}[/tex]
[tex]\mathbf{1000 = 500t}[/tex]
Divide both sides by 500
[tex]\mathbf{2 = t}[/tex]
Rewrite as:
[tex]\mathbf{t = 2}[/tex]
Hence, it will take 2 hours to empty the tank at a rate of 500 gallons per minutes
Read more about variations at:
https://brainly.com/question/12009761
