Here is my proof, but I have received a lot of criticism for it from an engineer. I´ll show my proof first then the criticism. Please advise on where I went wrong (if at all)!
Let the apex of the cone be at the point $(-1,0,0)$ and the base lie in the plane $z = c$. Then the area of the base is $\pi(1-c^2)$ and the height of the cone is $c + 1$, so the volume of the cone is $$\frac{\pi(1-c)(1+c)^2}3$$If we take the derivative of this and set the result to 0, we arrive at the equation $$2(1-c)(1+c) - (1+c)^2 = 0 \implies 2(1-c) - (1+c) = 0 \implies c= \frac13$$The volume of the cone when $c = \frac13$ is $$\pi \cdot \frac23 \cdot \frac43 \cdot \frac43 = \frac{\frac{32}{27}\pi}3$$Dividing this by $\frac43 \pi$ (the volume of the sphere) gives $$\frac{32/27}{4} = \frac8{27} \approx 0.296 < 0.3$$
The Engineer says that $\frac43 \pi$ is not the volume of the sphere, since that volume would be correct if the sphere had a radius of 1 and he says it is very unclear that my sphere has a radius of 1. ud.
He also says that the height of the cone cannot be c + 1.
I don´t see anything wrong with my solution.