Answer
$a=\dfrac{18}{125}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use $
a=k\cdot\dfrac{mn^2}{y^3}
$ and solve for the value of $k$ with the given $
a,m,n
$ and $
y
$ values. Then use the equation of variation to solve for the value of the unknown variable.
$\bf{\text{Solution Details:}}$
Since $a$ is directly proportional to $m$ and $n^2$ and inversely proportional to $y^3,$ then $
a=k\cdot\dfrac{mn^2}{y^3}
.$ Substituting the given values, $
a=9, m=4,n=9
$ and $
y=3
,$ then the value of $k$ is
\begin{array}{l}\require{cancel}
a=k\cdot\dfrac{mn^2}{y^3}
\\\\
9=k\cdot\dfrac{(4)(9)^2}{(3)^3}
\\\\
9=k\cdot\dfrac{(4)(81)}{27}
\\\\
9=k\cdot\dfrac{(4)(\cancel{81}(3))}{\cancel{27}}
\\\\
9=k\cdot12
\\\\
\dfrac{9}{12}=k
\\\\
k=\dfrac{3}{4}
.\end{array}
Hence, the equation of variation is given by
\begin{array}{l}\require{cancel}
a=k\cdot\dfrac{mn^2}{y^3}
\\\\
a=\dfrac{3}{4}\cdot\dfrac{mn^2}{y^3}
\\\\
a=\dfrac{3mn^2}{4y^3}
.\end{array}
If $m=6,n=2,$ and $y=5,$ then
\begin{array}{l}\require{cancel}
a=\dfrac{3(6)(2)^2}{4(5)^3}
\\\\
a=\dfrac{3(6)(4)}{4(125)}
\\\\
a=\dfrac{3(6)(\cancel4)}{\cancel4(125)}
\\\\
a=\dfrac{18}{125}
.\end{array}
