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Mathematics

A sphere and a cone have equal volumes and equal radii. The ratio between the radius and height of the cone is :

  1. 3 : 4

  2. 4 : 3

  3. 4 : 1

  4. 1 : 4

Mensuration

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Answer

Let radius of cone and sphere be r cm and height of cone be h cm.

Given,

A cone and a sphere have equal volumes and their radius are equal.

∴ Volume of cone = Volume of sphere

⇒13πr2h=43πr3⇒h=4×3πr33πr2⇒h=4r⇒rh=14⇒r:h=1:4.\Rightarrow \dfrac{1}{3}πr^2h = \dfrac{4}{3}πr^3 \\[1em] \Rightarrow h = \dfrac{4 \times 3πr^3}{3πr^2} \\[1em] \Rightarrow h = 4r \\[1em] \Rightarrow \dfrac{r}{h} = \dfrac{1}{4} \\[1em] \Rightarrow r : h = 1 : 4.

Hence, Option 4 is the correct option.

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