KJSEA 2025 Maths Q30 — Volume & Displacement (Rise in Water Level)

KJSEA 2025 Grade 9 Measurement

Published

The Question

“A cylindrical tin of radius 1.8 cm contains water. A spherical ball of radius 1.5 cm is fully immersed in the water. Determine the rise in the water level, correct to one decimal place.”

1

Understand what displacement means

When the ball is fully submerged, it takes up space that water used to be able to fill, so the water has nowhere to go but up. The extra volume of water pushed upward is exactly equal to the volume of the ball. That risen water forms a short cylinder of water sitting on top of the original level, with the same base as the tin.

Fully immersed is the key phrase — the whole ball is under water, so the whole ball volume counts.

2

Find the volume of the ball

The ball is a sphere, so use the sphere volume formula with radius 1.5 cm. Cubing 1.5 gives 3.375, and four-thirds of that is 4.5, so the volume is four-and-a-half pi cubic centimetres. Leaving pi in the answer keeps the numbers exact and makes the next step easier.

Vball=43πr3=43π(1.5)3V_{\text{ball}} = \frac{4}{3}\pi r^{3} = \frac{4}{3}\pi (1.5)^{3}
=43π(3.375)=4.5π cm3= \frac{4}{3}\pi (3.375) = 4.5\pi \ \text{cm}^{3}
3

Write the volume of the risen water

The water rises across the full circular base of the tin, so the extra water is a cylinder with the tin's radius of 1.8 cm and an unknown height h, the rise we want. Squaring 1.8 gives 3.24, so this risen cylinder of water has volume 3.24 pi times h.

Vrise=πR2h=π(1.8)2h=3.24πhV_{\text{rise}} = \pi R^{2} h = \pi (1.8)^{2} h = 3.24\pi h
4

Set the two volumes equal and solve

Because the displaced water equals the ball's volume, set the two expressions equal. Pi appears on both sides, so it cancels out, leaving a simple division. Dividing 4.5 by 3.24 gives about 1.39, which rounds to 1.4 cm to one decimal place.

4.5π=3.24πh4.5\pi = 3.24\pi h
h=4.53.24=1.388…h = \frac{4.5}{3.24} = 1.388\ldots
h≈1.4 cmh \approx 1.4 \ \text{cm}

Final Result

The water level rises by about 1.4 cm, correct to one decimal place.

Why this method works

The method works because a solid that is fully submerged cannot compress the water — it simply occupies space, forcing an equal volume of water to move elsewhere. Since the tin is a straight-sided cylinder, that displaced water spreads evenly over the circular base and stacks up as a new cylinder of the same radius. Equating the ball's volume to this cylinder of risen water turns a physical idea, displacement, into a single equation, and because both sides share the factor pi it cancels cleanly to leave the height directly.

✓

Reverse it: a 1.4 cm rise gives 3.24 pi times 1.4, about 4.54 pi, which matches the ball's 4.5 pi (the tiny gap is just the rounding of h).