Question

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

Assertion (A): Number of cylinders formed \[=\frac{1}{3} \times 3 \times 3 \times 3\]

Reason (R): Number of cylinders formed \[=\frac{\frac{1}{3} \pi(3)^{2} \times 3}{\pi(1)^{2} \times 1}\]

Options

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A.

• Both A and R are true and R is the incorrect reason for A.

Answer 1

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

Explanation:

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 \[= \frac{1}{3}\, \pi R^2 H\]

Volume of cylinder \[= \pi r^2 h.\]

Given,

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

∴ Volume of cone = n × Volume of cylinder

\[\Rightarrow \frac{1}{3}\, \pi R^2 H = n \times \pi r^2 h\]

\[\Rightarrow \frac{1}{3}\, \pi (3)^2 \times 3 = n \times \pi (1)^2 \times 1\]

\[\Rightarrow \frac{1}{3}\, (3)^2 \times 3 = n \times 1 \times 1\]

\[\Rightarrow n = \frac{1}{3} \times 3 \times 3 \times 3\]

⇒ n = 9

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