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A cone is 8.4 cm high and the radius of its base is 2.1 cm. It is melted and recast into a sphere. Find the radius of the sphere.

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Answer

Radius of base of cone (r) = 2.1 cm and height (h) = 8.4 cm.

Let radius of sphere be R cm.

Since, cone is recasted into sphere hence, their volume will be equal.

Volume of cone = Volume of sphere.

∴13πr2h=43πR3\therefore \dfrac{1}{3}πr^2h = \dfrac{4}{3}πR^3

Multiplying both sides by 3π\dfrac{3}{π}

⇒r2h=4R3⇒R3=r2h4⇒R3=(2.1)2×8.44⇒R3=(2.1)2×(2.1)⇒R3=(2.1)3⇒R=2.1 cm.\Rightarrow r^2h = 4R^3 \\[1em] \Rightarrow R^3 = \dfrac{r^2h}{4} \\[1em] \Rightarrow R^3 = \dfrac{(2.1)^2 \times 8.4}{4} \\[1em] \Rightarrow R^3 = (2.1)^2 \times (2.1) \\[1em] \Rightarrow R^3 = (2.1)^3 \\[1em] \Rightarrow R = 2.1 \text{ cm}.

Hence, the radius of the sphere = 2.1 cm.

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