Directions : Read the following passage carefully and answer the questions…
2022
Directions : Read the following passage carefully and answer the questions given below.
A hollow cylinder A of radius 14cm is full with water and a solid cylinder B with radius 7cm is put inside the hollow cylinder, then the remaining quantity of water is 9240 cm³. The height of both the cylinder is same.
If the radius of cylinder B is increased by 3.5 cm, then find the excessive amount of water that would have spilled from the cylinder A.
- A.
3050 cm³
- B.
4870 cm³
- C.
3800 cm³
- D.
3850 cm³
- E.
4550 cm³
Show answer & explanation
Correct answer: D
Concept
The volume of a cylinder is V = π r2 h, where r is the radius and h the height. When a solid cylinder is fully submerged in a vessel that is already full, it displaces — and therefore spills — a volume of water equal to its own volume. If the submerged cylinder's radius grows while its height stays the same, the ADDITIONAL water that spills equals the INCREASE in its volume, i.e. π h (r₂2 − r₁2), where r₂ is the new radius and r₁ the old. The increase depends only on the change in r2, not on the outer vessel's radius.
Application
Find the common height h from the first situation. Water left after inserting B = volume of A − volume of B = π h (R2 − r2) with R = 14, r = 7.
So π h (142 − 72) = 9240 → (22/7) × h × (196 − 49) = 9240 → (22/7) × h × 147 = 9240.
(22/7) × 147 = 22 × 21 = 462, so 462 h = 9240 → h = 20 cm.
New radius of B = 7 + 3.5 = 10.5 cm. Extra water spilled = increase in B's volume = π h (10.52 − 72).
= (22/7) × 20 × (110.25 − 49) = (22/7) × 20 × 61.25 = 3850 cm3.
Cross-check
Compute the two B-volumes separately: old B = (22/7) × 49 × 20 = 3080 cm3; new B = (22/7) × 110.25 × 20 = 6930 cm3. Their difference 6930 − 3080 = 3850 cm3, which matches. Also 10.5 cm < 14 cm, so the larger B still fits inside A.