A hollow sphere of internal and external diameters $4cm$ and $8cm$ respectively is melted into a cone of base diameter $8cm$. Find the height of the cone.
Last updated date: 19th Mar 2023
•
Total views: 309.3k
•
Views today: 3.91k
Answer
309.3k+ views
Hint: - When we melt one shape into another shape then volume of both the shapes are same (ideal conditions)
Given:
Diameter of the cone is equal to $8cm$.
So the radius ${r_1}$of the cone$ = \dfrac{{{\text{diameter}}}}{2} = \dfrac{8}{2} = 4cm$
As we know the volume of the cone is $\dfrac{1}{3}\pi r_1^2h$, where $h$is the height and ${r_1}$is the radius of the cone respectively.
And we know that Volume of hollow sphere of outer radius$\left( R \right)$and inner radius$\left( r \right)$$ = \dfrac{4}{3}\pi \left( {{R^3} - {r^3}} \right)$
Outer radius$\left( R \right) = 4cm$and inner radius $\left( r \right) = 2cm$
According to given condition
Volume of resulting cone = volume of hollow sphere
$
\dfrac{1}{3}\pi r_1^2h = \dfrac{4}{3}\pi \left( {{R^3} - {r^3}} \right) \\
\Rightarrow r_1^2h = 4\left( {{R^3} - {r^3}} \right) \\
\Rightarrow {4^2}h = 4\left( {{4^3} - {2^3}} \right) \\
\Rightarrow 4h = 64 - 8 = 56 \\
\Rightarrow h = \dfrac{{56}}{4} = 14cm \\
$
Hence, the height of the cone is$14cm$.
Note: -In such types of questions always remember the formula of volume of hollow sphere with inner and outer radius and the formula of volume of cone, then according to given condition both volume are equal then substitute the given values we will get the required value of the height of the cone.
Given:
Diameter of the cone is equal to $8cm$.
So the radius ${r_1}$of the cone$ = \dfrac{{{\text{diameter}}}}{2} = \dfrac{8}{2} = 4cm$
As we know the volume of the cone is $\dfrac{1}{3}\pi r_1^2h$, where $h$is the height and ${r_1}$is the radius of the cone respectively.
And we know that Volume of hollow sphere of outer radius$\left( R \right)$and inner radius$\left( r \right)$$ = \dfrac{4}{3}\pi \left( {{R^3} - {r^3}} \right)$
Outer radius$\left( R \right) = 4cm$and inner radius $\left( r \right) = 2cm$
According to given condition
Volume of resulting cone = volume of hollow sphere
$
\dfrac{1}{3}\pi r_1^2h = \dfrac{4}{3}\pi \left( {{R^3} - {r^3}} \right) \\
\Rightarrow r_1^2h = 4\left( {{R^3} - {r^3}} \right) \\
\Rightarrow {4^2}h = 4\left( {{4^3} - {2^3}} \right) \\
\Rightarrow 4h = 64 - 8 = 56 \\
\Rightarrow h = \dfrac{{56}}{4} = 14cm \\
$
Hence, the height of the cone is$14cm$.
Note: -In such types of questions always remember the formula of volume of hollow sphere with inner and outer radius and the formula of volume of cone, then according to given condition both volume are equal then substitute the given values we will get the required value of the height of the cone.
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Name the Largest and the Smallest Cell in the Human Body ?

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

A ball impinges directly on a similar ball at rest class 11 physics CBSE

Lysosomes are known as suicidal bags of cell why class 11 biology CBSE

Two balls are dropped from different heights at different class 11 physics CBSE

A 30 solution of H2O2 is marketed as 100 volume hydrogen class 11 chemistry JEE_Main

A sample of an ideal gas is expanded from 1dm3 to 3dm3 class 11 chemistry CBSE
