Question

Two identical cones of height 2 cm are placed one below another vertically, pointing their apex downwards. Upper cone is filled with water. Water from upper cone drops into lower cone through hole in the upper cone. What will be the height of water in the lower cone when height of the water in upper cone is half ?

Answer 1

Let R be the radius of the base of each come and H (H=2) the total height.

When the height of water in the top cone is 1, the radius of the free surface is r and we must have r/R=1/H so r=R/H. The volume of water that left the top cone is therefore V = (1/3) pi R^2 H - (1/3) (R/H)^2 1 or

V=(1/3) pi R^2 (H^3-1)/H^2

If at that time the height of the water in the lower cone is h, then the radius of the free surface is hR/H and therefore the volume of water is (1/3) pi (hR/H)^2 h = (1/3) pi R^2 h^3/H^2

Equating the two obtained volumes we see that h^3 = H^3 - 1 = 2^3 - 1 = 7 so h is the cube root of 7 and therefore the correct answer is (D)