A leak proof cylinder of length $1$ m, made of a metal which has a very low coefficient of expansion, is floating vertically in water at $0\ ^\circ\text{C}$ such that its height above the water surface is $20$ cm. When the temperature of water is increased to $4\ ^\circ\text{C}$, the height of the cylinder above the water surface becomes $21$ cm. The density of water at $T = 4\ ^\circ\text{C}$, relative to the density at $T = 0\ ^\circ\text{C}$, is close to:
Answer: (C) $1.01$
The weight of the cylinder equals the weight of water displaced, so $\rho\,A\,h_{\text{sub}}$ stays the same.
Submerged length: $80$ cm at $0\ ^\circ$C and $79$ cm at $4\ ^\circ$C.
$$\frac{\rho_4}{\rho_0} = \frac{80}{79} \approx 1.013 \approx 1.01$$
(The official answer key lists $1.04$.)
Solution by Sreeraj P, M.Sc Physics