A cylinder of radius 2 contains water to a depth ℎ. A cone with base radius and perpendicular height is lowered into the water. When the bottom of the cone is resting on the bottom of the beaker the water just covers the top of the cone. Find an expression for the initial depth of the water ℎ in terms of .
How could I find the new depth, so that I can define $h$ as $new depth - volume of cone$?
The volume of water is $V_w=\pi(2a)^2 h$. The volume of the cone is $V_c=\frac13\pi (a)^2 a$. To find the new height of the water $h^*$ use the equation: $$ \pi(2a)^2 h^*=V_w+V_c\implies h^*=\frac{V_w+V_c}{\pi(2a)^2}=h+\frac1{12}a. $$
Can you take it from here?