Answer
$y=\dfrac{45}{16}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use $
y=k\cdot\dfrac{x}{m^2r^2}
$ and solve for the value of $k$ with the given $
y,x,m
$ and $
r
$ values. Then use the equation of variation to solve for the value of the unknown variable.
$\bf{\text{Solution Details:}}$
Since $y$ varies directly as $x$ and inversely as $m^2$ and $r^2,$ then $
y=k\cdot\dfrac{x}{m^2r^2}
.$ Substituting the given values, $
y=\dfrac{5}{3},x=1,m=2,
$ and $
r=3
,$ then the value of $k$ is
\begin{array}{l}\require{cancel}
y=k\cdot\dfrac{x}{m^2r^2}
\\\\
\dfrac{5}{3}=k\cdot\dfrac{1}{(2)^2(3)^2}
\\\\
\dfrac{5}{3}=k\cdot\dfrac{1}{(4)(9)}
\\\\
\dfrac{5}{3}=k\cdot\dfrac{1}{36}
\\\\
36\left(\dfrac{5}{3}\right)=\left(k\cdot\dfrac{1}{36}\right)36
\\\\
\cancel{36}^{12}\left(\dfrac{5}{\cancel3}\right)=k
\\\\
12(5)=k
\\\\
k=60
.\end{array}
Hence, the equation of variation is given by
\begin{array}{l}\require{cancel}
y=k\cdot\dfrac{x}{m^2r^2}
\\\\
y=60\cdot\dfrac{x}{m^2r^2}
.\end{array}
If $x=3,m=1$ and $r=8,$ then
\begin{array}{l}\require{cancel}
y=60\cdot\dfrac{x}{m^2r^2}
\\\\
y=60\cdot\dfrac{3}{(1)^2(8)^2}
\\\\
y=60\cdot\dfrac{3}{(1)(64)}
\\\\
y=\cancel{60}^{15}\cdot\dfrac{3}{(1)(\cancel{64}^{16})}
\\\\
y=\dfrac{3(15)}{16}
\\\\
y=\dfrac{45}{16}
.\end{array}
