Question 1136494
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Let r be the radius of the cone.  With height 9, the slant height l is then {{{sqrt(r^2+81)}}}.<br>
The problem says that, ignoring units, the volume and lateral surface area are the same.  So<br>
{{{(1/3)(pi)(r^2)(h) = (pi)(r)(l)}}}
{{{(1/3)(pi)(r^2)(9) = (pi)(r)(sqrt(r^2+81))}}}
{{{3r = sqrt(r^2+81)}}}
{{{9r^2 = r^2+81}}}<br>
I'm not sure of what your definition of "vertical angle" is in this problem.  The angle between the base and the slant height of the cone is<br>
{{{arccos(r/sqrt(r^2+81)) = arccos(r/(3r)) = arccos(1/3)}}}<br>
which is 70.53 degrees, to 2 decimal places.