Answer
$r=\dfrac{12}{5}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use $
r=k\cdot\dfrac{m^2}{s}
$ and solve for the value of $k$ with the given $
r,m
$ and $
s
$ values. Then use the equation of variation to solve for the value of the unknown variable.
$\bf{\text{Solution Details:}}$
Since $r$ varies directly as the square of $m$ and inversely as $s,$ then $
r=k\cdot\dfrac{m^2}{s}
.$ Substituting the given values, $
r=12,m=6,
$ and $
s=4
,$ then the value of $k$ is
\begin{array}{l}\require{cancel}
r=k\cdot\dfrac{m^2}{s}
\\\\
12=k\cdot\dfrac{(6)^2}{4}
\\\\
12=k\cdot\dfrac{36}{4}
\\\\
12=9k
\\\\
\dfrac{12}{9}=k
\\\\
k=\dfrac{4}{3}
.\end{array}
Hence, the equation of variation is given by
\begin{array}{l}\require{cancel}
r=k\cdot\dfrac{m^2}{s}
\\\\
r=\dfrac{4}{3}\cdot\dfrac{m^2}{s}
\\\\
r=\dfrac{4m^2}{3s}
.\end{array}
If $m=6$ and $s=20,$ then
\begin{array}{l}\require{cancel}
r=\dfrac{4m^2}{3s}
\\\\
r=\dfrac{4(6)^2}{3(20)}
\\\\
r=\dfrac{4(36)}{3(20)}
\\\\
r=\dfrac{\cancel4(36)}{3(\cancel{20}^5)}
\\\\
r=\dfrac{36}{3(5)}
\\\\
r=\dfrac{\cancel{36}^{12}}{\cancel3(5)}
\\\\
r=\dfrac{12}{5}
.\end{array}
