Nub of the Matter

$V_{hb} =\text{Volume of a half bubble (hemisphere ) with radius r, }=\dfrac{2*\pi} 3 * r^3.\\ V_{cap}=\text{Volume of sphere ,radius R, inside the half bubble is the cap volume with height=h.}\\ V_{cap}=\dfrac{h^2*\pi} 3 * (3*R - h), \ we\ have\ R=1, \ and \ \color{#3D99F6}{h^2=2h - r^2...(**)}\\ Solving\ the\ quadratic\ \color{#3D99F6}{h=1 - \sqrt{1 - r^2}....(***) }.\\ Repeatedly\ \ using\ (**)\ \ V_{cap}=\dfrac \pi 3 *(2h - r^2)(3 - h)\\ =\dfrac \pi 3 *\{(6+r^2)*h -3r^2 - 2h^2\}=\dfrac \pi 3 *\{(6+r^2)*h -3r^2 - 2(2h-r^2) \}\\ =\dfrac \pi 3 *\{(2+r^2)*h - r^2\}=\dfrac \pi 3 *\{(2+r^2)*(1-\sqrt{1- r^2}) - r^2 \}....used..(***)\\ =\dfrac \pi 3 *\{(2+r^2) - (2+r^2) *\sqrt{1 - r^2}) - r^2\}=\dfrac \pi 3 *(2 - (2+ r^2) *\sqrt{1 - r^2}) \\ \therefore\ the \ required\ volume\ V= V_{hb} - V_{cap}.\\ V=\dfrac \pi 3*(2r^3 - 2+(2 +r^2) *\sqrt{1 - r^2})\\ V '=0,\ \implies\ \ 6r^2 - 2r*\sqrt{1 - r^2} + (2 +r^2)*r*\dfrac 1{ \sqrt{1 - r^2}} = 0,\\$ $Dividing\ by\ r\neq 0,\ and\ multiplying\ by\ \sqrt{1 - r^2},\\ 6r*{\sqrt{1 - r^2} - 2(1 - r^2) + (2 +r^2)} =0,\\ \implies\ \sqrt{1 - r^2}= - \dfrac{ 3r^2}{6r}= - \frac r 2,\ \ squaring\ both\ sides,\\ 1-r^2=\dfrac{r^2} 4.\ \ \therefore\ r=\sqrt{\dfrac 4 5}= \Large \ \ \ \ \color{#D61F06}{0.894427}$

Let the x-axis be the line joining the centers of the half bubble and sphere.

Let x = a be the point where the plane of intersection of the half bubble and sphere meets the x-axis.

Then a^2 + r^2 = 1

The volume H of the half bubble is 1/2(4/3 pi r^3) = 2/3 pi r^3 i.e. H/pi = 2/3 r^3

The volume of a circular disc of thickness dx and radius y is pi y^2 dx and the volume U of the sphere inside the half bubble is:

U = Integral ( from x = a to x = 1) of pi y^2 dx where x^2 + y^2 = 1

U/pi = Integral ( from x = a to x = 1) of (1 - x^2)dx = x - x^3/3 ( from x = a to x = 1) = 2/3 - (a - a^3/3)

Let V be the required volume between the inside of the half bubble and outside the sphere.

Then V = H - U, and V/pi = 2/3 r^3 - 2/3 + (a - a^3/3) = 2/3 r^3 - 2/3 + (1 - r^2)^(0.5) - [ (1 - r^2)^(3/2) ] / 3

For a maximum dV/dr = 0 that is 2 r^2 - r (1 - r^2)^(-0.5) + r (1 - r^2)^(0.5) = 0

2 r^2 (1 - r^2)^(0.5) - r + r(1 - r^2) = 0 ==> 2(1 - r^2)^(0.5) - r = 0 ==>4(1 - r^2) = r^2 ==> 5r^2 = 4

r = 2 / [ 5^(0.5) ] = 0.8944.