Question

A conical empty vessel is to be filled up completely by pouring water into it successively with the help of a cylindrical can of diameter 6cm and height 12cm. The radius of the conical vessel is 9cm and its height is 72cm. How many times will it require to pour water into the conical vessel to fill it completely, if in each time the cylindrical can is filled with water completely?
A) 14
B) 18
C) 20
D) 12

Answer 1

Hint: To find the number of times cylindrical can require to pour water into the conical vessel to fill it completely, we will use the concept of volume.

Complete step by step answer:
Let's suppose ‘n’ be the required number of times.
After filling the conical vessel by pouring water into it ‘n’ times with the help of cylindrical can, the volume of water will become equal.
Therefore, Volume of conical vessel = n (Volume of cylindrical can)
Since, Volume of conical vessel = $\dfrac{1}{3}\pi {r^2}h$
Volume of cylindrical can = $\pi {r^2}h$
Complete step by step solution:
Let ‘n’ be the required number of times it will require to fill it completely.
Then, the volume of the conical vessel will be equal to the volume of the cylindrical can.
It is given that -
Radius of conical vessel $\left( {{r_1}} \right) = 9cm$
Height of conical vessel$\left( {{h_1}} \right) = 72cm$
Radius of cylindrical can \[\left( {{r_2}} \right) = \dfrac{6}{2}cm = 3cm\]
Height of cylindrical can$\left( {{h_2}} \right) = 12cm$
We know that, Volume of the conical vessel $\left( {{V_1}} \right) = \dfrac{1}{3}\pi {r_1}^2{h_1}$
$\begin{gathered}
  {V_1} = \dfrac{1}{3}\pi \times {\left( 9 \right)^2} \times 72c{m^3} \\
  {V_1} = 24 \times 81\pi c{m^3} \\
\end{gathered} $
And the volume of cylindrical can $\left( {{V_2}} \right) = \pi {r_2}^2{h_2}$
$\begin{gathered}
  {V_2} = \pi \times {\left( 3 \right)^2} \times 12c{m^3} \\
  {V_2} = 9 \times 12\pi c{m^3} \\
\end{gathered} $
Therefore, ${V_1} = n\left( {{V_2}} \right)$
$\begin{gathered}
  24 \times 81\pi = n\left( {9 \times 12\pi } \right) \\
  n = \dfrac{{24 \times 81\pi }}{{9 \times 12\pi }} \\
  n = 18 \\
\end{gathered} $
Hence, the required number of times = 18
∴Option (B) is correct.

Note: The volumes of a cone and a cylinder that have identical bases and heights are proportional. If a cone and a cylinder have bases with equal areas, and both have identical heights, then the volume of the cone is one-third the volume of the cylinder.
The formula for the volume of a regular cone or right circular cone and the oblique cone is the same.
The slant height of the cone (l) (specifically right circular) is the distance from the vertex or apex to the point on the outer line of the circular base of the cone i.e. $l = \sqrt {{r^2} + {h^2}} $