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Mathematics

A cone and a sphere have equal volumes. If height of the cone = radius of the sphere = 10 cm, the radius of the cone is :

  1. 40 cm

  2. 10 cm

  3. 30 cm

  4. 20 cm

Mensuration

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Answer

Given,

Height of the cone (H) = radius of the sphere (r) = 10 cm

Volume of cone and sphere are equal.

Let radius of cone be R cm.

∴13πR2H=43πr3⇒R2H=4r3⇒R2=4r3H⇒R2=4×10310⇒R2=4×102⇒R2=400⇒R=400=20 cm.\therefore \dfrac{1}{3}πR^2H = \dfrac{4}{3}πr^3 \\[1em] \Rightarrow R^2H = 4r^3 \\[1em] \Rightarrow R^2 = \dfrac{4r^3}{H} \\[1em] \Rightarrow R^2 = \dfrac{4 \times 10^3}{10} \\[1em] \Rightarrow R^2 = 4 \times 10^2 \\[1em] \Rightarrow R^2 = 400 \\[1em] \Rightarrow R = \sqrt{400} = 20 \text{ cm}.

Hence, Option 4 is the correct option.

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