Question

A hollow sphere of internal and external diameters of 4 cm and 10 cm respectively, is melted into a cone of base diameter 5 cm. Find the height of the cone so formed.

Answer 1

Hint: Here, we need to find the height of the cone formed. We will first find out the internal and external volume of the sphere, and the volume of the cone. Using these, we will form an equation and solve it to find the height of the cone formed.

Formula used:
We will use the following formulas to solve the question.
The volume of a sphere, \[\dfrac{4}{3}\pi {r^3}\], where \[r\] is the radius of the sphere.
The volume of a cone, \[\dfrac{1}{3}\pi {r^2}h\], where \[h\] is the height of the cone and \[r\] is the radius of the base of the cone.

Complete step-by-step answer:
Let us assume the height of the cone to be \[h\] cm.

The radius of the base of the cone is half of its diameter, that is \[r = \dfrac{d}{2}\].
Substituting \[d = 5{\rm{ cm}}\] in the formula, we get
Radius of the base of the cone \[ = \dfrac{5}{2}\] cm.
Substituting \[d = 4{\rm{ cm}}\] in the formula, we get
Internal radius of the sphere \[ = \dfrac{4}{2} = 2\] cm.
Substituting \[d = 10{\rm{ cm}}\] in the formula, we get
External radius of the sphere \[ = \dfrac{{10}}{2} = 5\] cm.
Now, we need to find out the external and internal volume of the sphere.
We know that the formula of the volume of a sphere is \[\dfrac{4}{3}\pi {r^3}\].
Substituting \[r = 5\]cm in the formula, we get
External volume of the sphere \[ = \dfrac{4}{3}\pi {\left( 5 \right)^3}{\rm{ c}}{{\rm{m}}^3} = \dfrac{4}{3} \times \pi \times 125{\rm{ c}}{{\rm{m}}^3} = \dfrac{{500}}{3}\pi {\rm{ c}}{{\rm{m}}^3}\]
Substituting \[r = 2\]cm in the formula, we get
Internal volume of the sphere \[ = \dfrac{4}{3}\pi {\left( 2 \right)^3}{\rm{ c}}{{\rm{m}}^3} = \dfrac{4}{3} \times \pi \times 8{\rm{ c}}{{\rm{m}}^3} = \dfrac{{32}}{3}\pi {\rm{ c}}{{\rm{m}}^3}\]
The volume of the hollow sphere is equal to the difference in the external and internal volume of the sphere.
Therefore, we get
Volume of the hollow sphere \[ = \left( {\dfrac{{500}}{3}\pi - \dfrac{{32}}{3}\pi } \right){\rm{ c}}{{\rm{m}}^3} = \dfrac{{468}}{3}\pi {\rm{ c}}{{\rm{m}}^3}\]
Simplifying the expression, we get
Volume of the hollow sphere \[ = 156\pi {\rm{ c}}{{\rm{m}}^3}\]
Next, we will find the volume of the cone formed in terms of \[h\].
We know that the formula of the volume of a cone is \[\dfrac{1}{3}\pi {r^2}h\].
Substituting \[r = \dfrac{5}{2}\] cm in the formula, we get
Volume of the cone \[ = \dfrac{1}{3}\pi {\left( {\dfrac{5}{2}} \right)^2}h{\rm{ c}}{{\rm{m}}^3} = \dfrac{1}{3} \times \pi \times \dfrac{{25}}{4} \times h{\rm{ c}}{{\rm{m}}^3} = \dfrac{{25}}{{12}}\pi h{\rm{ c}}{{\rm{m}}^3}\]
Now, we will find the height of the cone.
The material from which the hollow sphere is made is used to make the cone.
Therefore, the volume of the hollow sphere is equal to the volume of the cone.
Thus, we get the equation
Volume of hollow sphere \[ = \] Volume of cone
Substituting volume of sphere as \[156\pi {\rm{ c}}{{\rm{m}}^3}\] and volume of the cone as \[\dfrac{{25}}{{12}}\pi h{\rm{ c}}{{\rm{m}}^3}\], we get
\[ \Rightarrow 156\pi = \dfrac{{25}}{{12}}\pi h\]
We will simplify this equation to get the value of \[h\].
Rewriting the equation, we get
\[ \Rightarrow \dfrac{{156\pi }}{{\dfrac{{25}}{{12}}\pi }} = h\]
Simplifying the equation, we get
\[\begin{array}{l} \Rightarrow \dfrac{{156}}{{\dfrac{{25}}{{12}}}} = h\\ \Rightarrow \dfrac{{156 \times 12}}{{25}} = h\end{array}\]
Thus, we get
\[ \Rightarrow h = 74.88\] cm
\[\therefore\] We get the height of the cone as \[74.88\] cm

Note: Here in this question, the volume of the hollow sphere will be equal to the volume of the cone because the material from which the hollow sphere is made is used to make the cone. Also, we have not substituted the value of \[\pi \] anywhere because in the end it will get cancelled. In this way, we can minimize the calculation.