Answer
$f=27$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use $
f=kg^2h
$ and solve for the value of $k$ with the given $
f,g
$ and $
h
$ values. Then use the equation of variation to solve for the value of the unknown variable.
$\bf{\text{Solution Details:}}$
Since $f$ varies jointly as $g^2$ and $h,$ then $
f=kg^2h
.$ Substituting the given values, $
f=50,g=5,
$ and $
h=4
,$ then the value of $k$ is
\begin{array}{l}\require{cancel}
f=kg^2h
\\\\
50=k(5)^2(4)
\\\\
50=k(25)(4)
\\\\
50=k(100)
\\\\
\dfrac{50}{100}=k
\\\\
k=\dfrac{\cancel{50}}{\cancel{50}\cdot2}
\\\\
k=\dfrac{1}{2}
.\end{array}
Hence, the equation of variation is given by
\begin{array}{l}\require{cancel}
f=kg^2h
\\\\
f=\dfrac{1}{2}g^2h
.\end{array}
If $g=3$ and $h=6,$ then
\begin{array}{l}\require{cancel}
f=\dfrac{1}{2}g^2h
\\\\
f=\dfrac{1}{2}(3)^2(6)
\\\\
f=\dfrac{1}{2}(9)(6)
\\\\
f=\dfrac{1}{\cancel2}(9)(\cancel2\cdot3)
\\\\
f=9(3)
\\\\
f=27
.\end{array}