# Sinking Feeling

Two vases are cylindrical in shape. Can you work out the original depth of the water in the larger vase?

## Problem

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Two vases are cylindrical in shape.

The larger vase has diameter 20 cm.

The smaller vase has diameter 10 cm and height 16 cm.

The larger vase is partially filled with water.

Then the empty smaller vase, with the open end at the top, is slowly pushed down into the water, which flows over its rim.

When the smaller vase is pushed right down, it is half full of water.

The larger vase is partially filled with water.

Then the empty smaller vase, with the open end at the top, is slowly pushed down into the water, which flows over its rim.

When the smaller vase is pushed right down, it is half full of water.

**What was the original depth of the water in the larger vase?**This problem is taken from the UKMT Mathematical Challenges.

## Student Solutions

The water completely fills the space between the cylinders up to the height of the smaller cylinder, and also half of the smaller cylinder. The volume of a cylinder is $V=\pi r^2h$, so can calculate the total volume of water by adding these two volumes together.

Between the cylinders the volume is:

$$V = \pi[10^2-5^2]\times 16=1200\pi$$

The volume in the smaller cylinder is

$$V = \pi (5^2 \times 8)=200\pi$$

So the total volume is:

$$V= 1200\pi + 200\pi = 1400\pi$$

In the large cylinder without the smaller cylinder, this volume occupies up to a height $h$ that satisfies:

$$100h\pi = 1400\pi \Rightarrow h=14$$