For reference: A cone and a cylinder, both straight, have the same volume and identical bases. Knowing that both are inscribable in a sphere of radius R, what is the height H of the cone (as a function of R)?(A: $\frac{6R}{5}$)
I have not been able to demonstrate this relationship.
$r_{ci}=r_{co} = r\\ r_e=R\\ V_{ci}=V_{co}\implies \pi r^2.h_{ci}=\frac{1}{3}.\pi r^2.h_{ci}\\ \therefore h_{co}=3h_{ci}\\ V_e=\frac{4}{3}.\pi .R^3\\ R^2 = r^2+(h_{co}-R)^2\\ 4R^2=h_{ci}^2+4r^2\implies h_{ci}^2=4(R^2-r^2)\\ (\frac{h_{co}}{3})^2=4(R^2-r^2)\implies h_{co}^2=36(R^2-r^2)\\ h_{co} = 6\sqrt{R^2-r^2}$
I don't see how to continue or even if the question's relationship can exist
From your work, we subtract the second equation from the first one and then we use the third one: $$\begin{cases} R^2 = r^2+(h_{co}-R)^2\\ R^2=\frac{h_{ci}^2}{4}+r^2\\ h_{co}=3h_{ci} \end{cases}\implies h_{co}-R=\frac{h_{ci}}{2}=\frac{h_{co}}{6} \implies h_{co}=\frac{6R}{5}.$$