The volume of an ‘ice cream cone’ is that of the hemisphere, 2/3 πr3, plus that of the cone, πr2h/3. If the angle at the bottom of the cone is 30°, then we can make a right triangle with angle 15°. It’s height is r/tan(15°), or 3.372r. This makes the volume of the ice cream cone 5.372rπr3/3. This is 500ml, or 500cm3. We can then make the equation 5.372rπr2/3=500cm3, which solves to r=9.428. Now, the surface area of the hemisphere is 3πr2, and that of the cone, πr(r+√(h2+r2)). But we must subtract 2πr2 for the 2 circles that are not exposed. That leaves us with 2πr2 + πr√(h2+r2). Plug in r=9.428 and h=3.372r to get our answer. 1601cm2.
V=500cm^3=1.791\pi r^2\\[8pt] r=9.428cm\\[16pt] SA=2\pi r^2+\pi r\sqrt{h^2+r^2}\\[8pt] SA=2\pi r^2+\pi r\sqrt{1.07177r^2}\\[8pt] SA=5.732\pi r^2\\[8pt] SA=1601cm^2