A ball of diameter $4$ cm is kept on top of a hollow cylinder standing vertically. The height of the cylinder is $3$ cm, while its volume is $9 \pi$ cm$^3$. Then the vertical distance, in cm, of the topmost point of the ball from the base of the cylinder is

We have a ball sitting on top of a hollow cylinder.

We have a hollow cylinder with height = 3 cm and volume = $9\pi$ cm³, and a ball with diameter = 4 cm (so radius = 2 cm). The ball is placed on top of the cylinder.

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Since we know the volume of the cylinder, we can find its radius using the volume formula.

Volume of cylinder = $\pi r^2 h$

Given: Volume = $9\pi$ cm³ and height = 3 cm

$9\pi = \pi \times r^2 \times 3$

$9\pi = 3\pi r^2$

$9 = 3r^2$

$r^2 = 3$

$r = \sqrt{3}$ cm

So the cylinder has radius $\sqrt{3} \approx 1.73$ cm.

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Here's the key insight: Since the ball has radius 2 cm and the cylinder has radius $\sqrt{3} \approx 1.73$ cm, the ball is larger than the cylinder opening!

This means the ball doesn't just sit flat on top of the cylinder. Instead, it sinks partially into the cylinder opening, resting on the circular rim at the top.

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When a sphere of radius $R$ rests on a circular rim of radius $r$ (where $R > r$), the center of the sphere is at height $\sqrt{R^2 - r^2}$ above the rim.

In our case, ball radius $R = 2$ cm and cylinder radius $r = \sqrt{3}$ cm.

Height of ball center above cylinder rim = $\sqrt{R^2 - r^2}$

$= \sqrt{2^2 - (\sqrt{3})^2}$

$= \sqrt{4 - 3}$

$= \sqrt{1}$

$= 1$ cm

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The topmost point of the ball is at:

Height of cylinder base to rim: 3 cm

Height of ball center above rim: 1 cm

Height from ball center to topmost point: 2 cm (radius of ball)

Total height = $3 + 1 + 2 = 6$ cm

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When a sphere sits on a circular opening smaller than itself, it doesn't rest on the surface - it sinks in until it touches the rim. The center height above the rim follows the formula $\sqrt{R^2 - r^2}$, which comes from the Pythagorean theorem applied to the right triangle formed by the sphere center, rim contact point, and the point directly below the center.

Answer: 6 cm