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Finding the relationship between the volume of the cylinder and volume of a cone.

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Last updated date: 20th Jun 2024
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Answer
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Hint: In this type of question, students should use the formula for the volume of the cylinder and the volume of the cone. It is necessary to apply the same radius and height in both the solids so that the volume of the cylinder and the cone can be related. A mathematical relationship can be developed only when the basic parameters of cone and cylinder are the same.

Complete step-by-step answer:
Let us assume that the radius of the cylinder and cone is given by $ r $ .
Also, let $ h $ be the height of the cylinder and the cone.
Let us assume that $ {V_{cylinder}} $ and $ {V_{cone}} $ denote the volume of the cylinder and the volume of the cone.
We know that the volume of a cylinder can be given as the product of $ \pi $ , square of the radius of the cylinder and height of the cylinder. This can be mathematically represented as $ \Rightarrow {V_{cylinder}} = \pi {r^2}h $ .
We know that the volume of a cone can be given as the product of $ \dfrac{1}{3} $ , $ \pi $ , square of the radius of the cylinder and height of the cone. This can be mathematically represented as
 $ \Rightarrow {V_{cone}} = \dfrac{1}{3}\pi {r^2}h $ .
Now, in order to find the relationship between the volume of the cylinder and volume of the cone, we should substitute $ \pi {r^2}h = {V_{cylinder}} $ in the formula $ {V_{cone}} = \dfrac{1}{3}\pi {r^2}h $ .
After substituting, we get, $ {V_{cone}} = \dfrac{1}{3} \cdot {V_{cylinder}} $
Therefore, the relationship between the volume of the cylinder and volume of the cone is $ \Rightarrow {V_{cone}} = \dfrac{1}{3} \cdot {V_{cylinder}} $ .
So, the correct answer is “ $ {V_{cone}} = \dfrac{1}{3} \cdot {V_{cylinder}} $ ”.

Note: In order to solve this type of questions, students should remember the formula for the volume of different three dimensional solids. Students must take note that for finding relationships between two quantities, the parameters on which the quantities depend upon should be the same. For instance, to find the relationship between area of sphere and area of hemisphere, the radius of both sphere and hemisphere should be equal.