Question

A person looking through a telescope T just sees the point A on the rim at the bottom of a cylindrical vessel
when the vessel is empty. When the vessel is completely filled with a liquid $(\mu =1.5),$ he observes a mark at the
centre, B, of the bottom without moving telescope or the vessel.
What is the height of the vessel if the diameter of its cross-section is 10 cm?

• A. 5.5 cm
• B. 6.6 cm
• C. 7.5 cm
• D. 8.45 cm

Answer 1

The correct option is D 8.45 cm

It is mentioned in the problem that on filling the vessel with the liquid, point B is observed for the same setting; this means that the image of point B, is observed at A, because of refraction of the ray at C. For refraction at C, For refraction at C,
$\frac{sinr}{sini}=a{\mu }_1=1.5$
Now,
$Sinr=\frac{AD}{AC}=\frac{10}{\sqrt{{10}^2+h^2}}$
Where h is the height of vessel.
$Sini=\frac{BD}{BC}=\frac{5}{\sqrt{5^2+h^2}}\therefore \frac{10}{\sqrt{{10}^2+h^2}}.\frac{\sqrt{{25}^2+h^2}}{5}=1.5\therefore \frac{25+h^2}{100+h^2}=\frac{9}{16}\therefore 400+16h^2=900+9h^2\therefore 7h^2=500\therefore h=8.45cm$