A cylindrical tank with radius $r$ meters and height $h$ meters is filled with water. A solid metallic cylinder of radius $\frac{r}{2}$ and height $2h$ is submerged completely in the tank. By what fraction does the water level in the tank rise?
$\frac{1}{2}$
Volume of submerged cylinder: $V = \pi \times \left(\frac{r}{2}\right)^2 \times (2h)$
$V = \pi \times \left(\frac{r^2}{4}\right) \times 2h = \frac{\pi \times r^2 \times h}{2}$.
Volume of water displaced = Base area of tank $\times$ Rise in water level
$\frac{\pi \times r^2 \times h}{2} = (\pi \times r^2) \times \text{Rise}$.
So the rise in water level = $\frac{h}{2}$.
Fractional rise = $\frac{\frac{h}{2}}{h} = \frac{1}{2}$.
Hence, $\frac{1}{2}$.
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