Question

r varies directly as s and inversely as t cubed. If r= 607.5 when s= 12 and r= 2, find r when s=42 and c= 6

Answer 1

$\bf \qquad \qquad \textit{double proportional variation}\\\\ \begin{array}{llll} \textit{\underline{y} varies directly with \underline{x}}\\ \textit{and inversely with \underline{z}} \end{array}\implies y=\cfrac{kx}{z}\impliedby \begin{array}{llll} k=constant\ of\\ \qquad variation \end{array}\\\\ -------------------------------\\\\ \textit{\underline{r} varies directly as \underline{s} and inversely as \underline{t} cubed}\qquad r=\cfrac{ks}{t^3}$

$\bf \textit{we also know that } \begin{cases} r=607.5\\ s=12\\ t=2 \end{cases}\implies 607.5=\cfrac{k12}{2^3} \\\\\\ 607.5=\cfrac{12k}{8}\implies 607.5=\cfrac{3k}{2}\implies \cfrac{607.5\cdot 2}{3}=k\implies 405=k \\\\\\ therefore\qquad \boxed{r=\cfrac{405s}{t^3}} \\\\\\ \textit{now, when s = 42 and t = 6, what is \underline{r}?}\qquad r=\cfrac{405(42)}{6^3}$