Question 1156779
<br>
{{{V = (1/3)(pi)(r^2)(h)}}}<br>
{{{A = (pi)(r)(l)}}} where l is the slant height.<br>
The volume and the lateral surface area are numerically equal:<br>
{{{(1/3)(pi)(r^2)(12) = (pi)(r)(l)}}}
{{{4r = l}}}<br>
The radius of the cone r and the slant height 4r form a right triangle with the altitude of the cone.  The angle in that triangle at the vertex of the cone is half the vertical angle of the cone; the sine of that angle is r/(4r) = 1/4.<br>
The vertical angle of the cone is<br>
{{{2*sin^(-1)(1/4)}}} = approximately 28.955 degrees.<br>