A geometry problem by vishwathiga jayasankar

The formula for the curved surface or lateral area of a cone is $A=\dfrac{1}{2}cL$ where $c$ is the circumference of the base and $L$ is the slant height. So by pythagorean theorem, compute for $L$ in terms of $r$ (radius) then substitute to the formula. You will come up with an equation to the fourth degree, then solve for $r$ . After that, use the formula $V=\dfrac{1}{3} A_bh$ where $A_b$ is the area of the base and $h$ is the height: to find for the volume.