Hemispherical Duck Pond

Let the center of the full hemispheric pond be the origin $O(0,0)$ of the $xy$ -plane. We can use a horizontal line through $O$ on the water surface of a full pond to be the $y$ -axis while the vertical line through $O$ be the $x$ -axis. Then the volume of the pond water with a depth of $D$ is given by:

$\begin{aligned} V(D) & = \int_{R-D}^R \pi {\color{#3D99F6}y^2} \ dx & \small \color{#3D99F6} \text{Since }x^2 + y^2 = R^2 \\ & = \int_{R-D}^R \pi {\color{#3D99F6}\left(R^2-x^2\right)} \ dx \\ & = \pi \left(R^2x - \frac {x^3}3\right) \bigg|_{R-D}^R \\ & = \frac {\pi D^2(3R-D)}3 \end{aligned}$

When the pond is full $V(R) = \dfrac {\pi R^2(3R-R)}3 = \dfrac 23 \pi R^3$ . When the pond is half-full $V\left(\frac R2\right) = \dfrac {\frac {R^2}4\left(3R-\frac R2\right)}3 = \dfrac 5{24}\pi R^3$ . Therefore, $\dfrac {V\left(\frac R2\right)}{V(R)} = \dfrac {\frac 5{24}}{\frac 23} = \boxed{\dfrac 5{16}}$ .