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Mathematics

A solid cone of height 3 cm and radius 3 cm is recast into solid cylinder each of height 1 cm and radius 1 cm.

Assertion(A): Number of cylinders formed = 13\dfrac{1}{3} x 3 x 3 x 3

Reason(R): Number of cylinders formed = 13π(3)2×3π(1)2×1\dfrac{\dfrac{1}{3}π(3)^2 \times 3}{π(1)^2 \times 1}

  1. A is true, R is false.

  2. A is false, R is true.

  3. Both A and R are true and R is correct reason for A.

  4. Both A and R are true and R is incorrect reason for A.

Mensuration

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Answer

Given, radius of solid cone, R = 3 cm

Height of solid cone, H = 3 cm

Radius of solid cylinder, r = 1 cm

Height of solid cylinder, h = 1 cm

By formula,

Volume of cone = 13πR2H\dfrac{1}{3}πR^2H

Volume of cylinder = πr2hπr^2h.

Given,

Solid cone is recasted into cylinders. Let no. of cylinders formed be n.

∴ Volume of cone = n × Volume of cylinder

⇒13πR2H=n×πr2h⇒13π(3)2×3=n×π(1)2×1⇒13(3)2×3=n×1×1⇒n=13×3×3×3⇒n=9.\Rightarrow \dfrac{1}{3}πR^2H = n × πr^2h\\[1em] \Rightarrow \dfrac{1}{3}π(3)^2 \times 3 = n × π(1)^2 \times 1\\[1em] \Rightarrow \dfrac{1}{3} (3)^2 \times 3 = n × 1 \times 1\\[1em] \Rightarrow n = \dfrac{1}{3} \times 3 \times 3 \times 3\\[1em] \Rightarrow n = 9.

∴ Both A and R are true and R is correct reason for A.

Hence, option 3 is the correct option.

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