Question

Seven metal hollow hemispheres of internal and external diameters 20 cm and 24 cm respectively are melted and recast into one solid cone of radius 28 cm. Find the height of the cone.

Answer 1

Hint:
Here, we are required to find the height of the cone which has been made by melting and recasting of seven metal hollow hemispheres. Hence, the volume of the cone will be equal to the volume of the seven hollow hemispheres. We will find the volume of both of them and then, after equating, we will find the required height of the cone.

Formula Used:
We will use the following formulas:
1) Volume of a hollow hemisphere \[ = \dfrac{2}{3}\pi \left( {{R^3} - {r^3}} \right)\]
2) Volume of cone \[ = \dfrac{1}{3}\pi {r^2}h\]

Complete step by step solution:
According to the question, the external diameter of a hollow hemisphere is 24 cm.
Now, the radius is half of diameter. So,
External radius of one hollow hemisphere, \[R = \dfrac{{24}}{2} = 12{\rm{cm}}\]
Similarly,
The internal diameter of a hollow hemisphere is 20 cm. So,
Internal radius of one hollow hemisphere, \[r = \dfrac{{20}}{2} = 10{\rm{cm}}\]
Now, we will find the volume of a hollow hemisphere.
Substituting \[R = 12\] and \[r = 10\] in the formula volume of a hollow hemisphere \[ = \dfrac{2}{3}\pi \left( {{R^3} - {r^3}} \right)\], we get,
Volume of a hollow hemisphere\[ = \dfrac{2}{3}\pi \left( {{{12}^3} - {{10}^3}} \right)\]
\[ \Rightarrow \]Volume of a hollow hemisphere \[ = \dfrac{2}{3}\pi \left( {1728 - 1000} \right)\]
\[ \Rightarrow \]Volume of a hollow hemisphere \[ = \dfrac{2}{3}\pi \left( {728} \right){\rm{c}}{{\rm{m}}^3}\]………………………\[\left( 1 \right)\]
Now, we are given a solid cone of radius 28 cm
Here, radius, \[r = 28\] and height \[ = h\] in the formula volume of cone \[ = \dfrac{1}{3}\pi {r^2}h\], we get
Volume of cone\[ = \dfrac{1}{3}\pi {\left( {28} \right)^2}h\] …………………………..\[\left( 2 \right)\]
Now, according to the question, seven metal hollow hemispheres are melted and recast into one solid cone of radius 28 cm.
Hence, the volume of these 7 hollow hemispheres will be equal to the volume of the cone.
Therefore, Multiplying equation \[\left( 1 \right)\] by 7 and equating it to equation \[\left( 2 \right)\],we get,
\[7 \times \dfrac{2}{3}\pi \left( {728} \right) = \dfrac{1}{3}\pi {\left( {28} \right)^2}h\]
Now, cancelling out the same terms, we get
\[ \Rightarrow 7 \times 2\left( {728} \right) = {\left( {28} \right)^2}h\]
Hence, for the height of the cone,
\[ \Rightarrow h = \dfrac{{7 \times 2 \times 728}}{{28 \times 28}}\]
Simplifying the expression, we get
\[ \Rightarrow h = \dfrac{{364}}{{28}} = 13\]

Therefore, the height of the cone is 13 cm.

Note:
A hemisphere is a three-dimensional figure which is half the sphere. Hence, the formula of volume of hemisphere is also half the formula of volume of a sphere. But, in this question, we are talking about the hollow hemisphere. A hollow hemisphere always has two diameters. Now, in this question, seven hollow hemispheres were melted and reshaped into one solid cone. We should keep in mind that whenever a given shape is melted to form a new shape, their volume remains the same. Hence, we equated their volumes to find the required height of the cone.