Assertion (A): If two metallic right circular cones of equal height and base radii 3 cm and 4 cm are melted and recast into a solid sphere of radius 5 cm, then the cones are of height 20 cm each. Reason (R): If two right circular cones of same height h an

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Solution

Q.
Assertion (A): If two metallic right circular cones of equal height and base radii 3 cm and 4 cm are melted and recast into a solid sphere of radius 5 cm, then the cones are of height 20 cm each.

Reason (R): If two right circular cones of same height h and base radii $r_{1}, r_{2}$ are melted and recast into a solid sphere of radius r, then $h = \frac{4 r^{2}}{r_{1} + r_{2}} .$

1 Both A and R are true, and R is the correct explanation of A.

2 Both A and R are true, but R is not the correct explanation of A.

3 A is true, but R is false.

4 A is false, but R is true

Ans.
(3)

When one solid shape is converted into another, the volume remains the same.

Sum of the volumes of two cones = volume of the sphere

$\Rightarrow \frac{1}{3} π r_{1}^{2} h + \frac{1}{3} π r_{2}^{2} h = \frac{4}{3} π r^{3} \Rightarrow h (r_{1}^{2} + r_{2}^{2}) = 4 r^{3} \Rightarrow h = \frac{4 r^{3}}{r_{1}^{2} + r_{2}^{2}} . . . (i) S o, R e a s o n (R) i s f a l s e . P u t t i n g r = 5, r_{1} = 3, r_{2} = 4 i n (i), w e o b t a i n h = \frac{4 \times 125}{25} = 20 cm . S o, A s s e r t i o n (A) i s t r u e .$

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