Question 1012281
We have to solve for the radius of the cylinder
IF we are not expected to be good enough at algebra.
That must be the case, since we are given the volume as encouragement to work hard ans solve for the radius og the cylinder.
However, the volume and the radius of the cylinder are not needed to solve the  problem.
For a cylinder of radius {{{R}}} and height {{{H}}} ,
{{{volume=pi*R^2*H}}} .
For Henry's cylinder, of radius {{{x}}} and height {{{7}}} (both in cm),
{{{volume=pi*x^2*7}}} , in cubic centimeters.
For a cone of radius {{{R}}} and height {{{H}}} ,
{{{volume=(1/3)*pi*R^2*H}}} .
For Henry's cone, of radius {{{2x}}} , height {{{h}}} (both in cm), and {{{volume=pi*x^2*7}}} ,  in cubic centimeters,
{{{(1/3)*pi*(2x)^2*H=pi*x^2*7}}}
{{{(1/3)*4x^2*H=x^2*7}}}
{{{(1/3)*4H=7}}}
{{{(4/3)*H=7}}}
{{{(3/4)*(4/3)*H=(3/4)*7}}}
{{{H=(3/4)*7}}} --> {{{highlight(H=21/4=5.25)}}} .
So, assuming the cup of water filled the cone completely, without any spillover,
the height of Henry's cone is {{{highlight(5.25cm)}}} .