1. Determine the value of the radius of flotation \(R_1\)

From the equation of a circle with center at point z=R, we get:

\begin{aligned} &r^2+(z-R)^2=R^2\\ &r(z)=\sqrt{R^2-(z-R)^2}=\sqrt{2 R \cdot z-z^2}\\ &R_1=r(h)=\sqrt{2 R \cdot h-h^2}=1.3 m \end{aligned}

2. Determine the value of the moment of inertia of the flotation surface

\begin{aligned} I_x=\frac{\pi \cdot R_1^4}{4}=2.243 \mathrm{~m}^4 \end{aligned}

3. Determine the volume of the submerged part of the floating body

There is a formula for the volume of a spherical segment, but personally I'm not a big fan of memorizing it:

\begin{aligned} V_z=\frac{\pi \cdot h^2}{3}(3 R-h)=2.209 \mathrm{~m}^3 \end{aligned}

An alternative is to calculate this volume using an integral formula for rotational figures:

\begin{aligned} V_z=\pi \int_0^h 2 R \cdot z-z^2 \mathrm{~d} z=2.209 \mathrm{~m}^3 \end{aligned}

As you can see, the final result is the same :-)

4. Determine the density of the material from which the hemisphere is made

From Archimedes' principle and the equilibrium condition

\begin{aligned} & G=W \\ & \rho_m \cdot g \cdot \frac{1}{2} \frac{4}{3} \pi \cdot R^3=\rho_w \cdot g \cdot V_z \\ & \rho_m=\frac{\rho_w \cdot g \cdot V_z}{g \cdot \frac{1}{2} \frac{4}{3} \pi \cdot R^3}=312.5 \frac{\mathrm{kg}}{\mathrm{m}^3} \end{aligned}