Question

If a cone and a sphere have equal radii and equal volumes. What is the ratio of the diameter of the sphere to the height of the cone?
A). \[1:1\]
B). \[1:2\]
C). \[2:1\]
D). \[3:2\]

Answer 1

Hint: Firstly let us consider that \[r\] be the radius of cone and sphere and \[h\] be the height of the cone and then apply the formula of volume of cone and sphere and then recall the formula for diameter to find out the ratio between diameter of sphere and height of cone and check which option is correct among them.

Complete step-by-step solution:
The space or capacity of a cone is defined by its volume. A cone is a three-dimensional geometric object with a circular base that tapers to a point called the apex or vertex from a flat base. A cone is made up of line segments, half-lines, or lines that connect a common point, the apex, to all the points on a base that are not in the same plane as the apex.A cone is made up of non-congruent circular discs placed on top of one another with the radius ratio of adjacent discs remaining constant. The sphere is a three-dimensional round solid shape in which all points on its surface are equally spaced from its centre. The fixed distance is known as the sphere's radius, and the fixed point is known as the sphere's centre. We will notice a change in shape as the circle is rotated. As a result of the rotation of the two-dimensional object known as a circle, the three-dimensional shape of a sphere is obtained.
Now according to the question:
Let us consider that \[r\] be the radius of the cone and sphere and \[h\] be the height of the cone.
Hence \[\text{Volume of cone = Volume of sphere}\]
As we know that \[\text{Volume of cone =}\dfrac{1}{3}\pi {{r}^{2}}h\] and \[\text{Volume of sphere}=\dfrac{4}{3}\pi {{r}^{3}}\] hence:
\[\Rightarrow \dfrac{1}{3}\pi {{r}^{2}}h=\dfrac{4}{3}\pi {{r}^{3}}\]
\[\Rightarrow h=\dfrac{4\times 3}{3}\times \dfrac{\pi {{r}^{3}}}{\pi {{r}^{2}}}\]
\[\Rightarrow h=4r\]
We can write this as:
\[\Rightarrow h=2r\times 2\]
\[\Rightarrow \dfrac{h}{2r}=2\]
As we know that diameter is equals to two times of the radius hence:
\[\Rightarrow \dfrac{h}{d}=2\]
\[\Rightarrow \dfrac{h}{d}=\dfrac{2}{1}\]
And we have to find out the ratio of diameter of sphere and height of cone that is \[\dfrac{d}{h}\] hence:
\[\Rightarrow \dfrac{d}{h}=\dfrac{1}{2}\]
Therefore the ratio between diameter of sphere and height of cone is \[1:2\]
Hence option \[(2)\] is correct.

Note: Another method for determining an object's volume is to totally submerge it in water and then measure the amount of water displaced by the object. The amount of water displaced is the object's volume. The volume of an object measured in cubic units such as cubic meter, cubic foot and many more.