Question

A right circular cone of diameter r cm and height 12 cm rests on the base of a right circular cylinder of radius r cm. Their bases are in the same plane and the cylinder is filled with water up to a height of 12cm. If the cone is removed, find the height to which water level falls.

Answer 1

Hint: In this problem, we are given a cone with given measurements resting on the base of a right circular cylinder whose bases are the same and the cylinder is filled with water up to 12 cm. We have to find the height of the water level when the cone is removed. We can find the volume of the cylinder and the volume of the cone and subtract the volume of the cone from the cylinder to get the height of the water when the cone is removed.

Complete step by step answer:
Here we are given a cone of r diameter and height 12cm rest on the base of a right circular cylinder of radius r cm whose bases are the same and the cylinder is filled with water up to 12 cm. We have to find the height of the water level when the cone is removed.

Radius of the cone = \[\dfrac{r}{2}=R\]cm.
Radius of cylinder = r cm.
Height of cone, h = 12cm.
We know that,
Volume of water left in the cylinder when cone is removed = Volume of cylinder – Volume of cone.
We know that volume of cylinder formula is \[\pi {{r}^{2}}h\] and the volume of cone is \[\dfrac{1}{3}\pi {{R}^{2}}h\].
Volume of water left in the cylinder when cone is removed = \[\pi {{r}^{2}}h-\dfrac{1}{3}\pi {{R}^{2}}h\].
We can now substitute the value of r and R in the above step, we get
\[\begin{align}
  & \Rightarrow \pi {{r}^{2}}\times 12-\dfrac{1}{3}\pi {{\left( \dfrac{r}{2} \right)}^{2}}\times 12 \\
 & \Rightarrow \pi {{r}^{2}}\times 12-\left( \pi {{r}^{2}} \right) \\
 & \Rightarrow \pi {{r}^{2}}\left( 12-1 \right) \\
 & \Rightarrow \pi {{r}^{2}}\times 11 \\
\end{align}\]
From the above step, we can see that the place of h in the cylinder formula is 11.
Therefore, the height of the water left in the cylinder after the cone is removed = 11cm.

Note: We should always remember the formulas in mensuration such as the volume of cylinder formula is \[\pi {{r}^{2}}h\] and the volume of cone is \[\dfrac{1}{3}\pi {{R}^{2}}h\]. Here we have differentiated the radius of the cone and the radius of the cylinder by using different variables for it.