Answer
(a) $x = 86.6~cm$
(b) $x = 64.7~cm$
(c) $x = 42.9~cm$
Work Step by Step
(a) If the tank is empty:
$\frac{50~cm}{x} = tan~30^{\circ}$
$x = \frac{50~cm}{tan~30^{\circ}}$
$x = 86.6~cm$
(b) If the tank is half full of water, we can find the horizontal distance a light ray travels from the top of the water to the eye:
$\frac{25~cm}{x_1} = tan~30^{\circ}$
$x_1 = \frac{25~cm}{tan~30^{\circ}}$
$x_1 = 43.3~cm$
We can find the angle of refraction:
$n_2~sin~\theta_2 = n_1~sin~\theta_1$
$sin~\theta_2 = \frac{n_1~sin~\theta_1}{n_2}$
$\theta_2 = sin^{-1}~(\frac{n_1~sin~\theta_1}{n_2})$
$\theta_2 = sin^{-1}~[\frac{(1.00)~sin~60^{\circ}}{1.33}]$
$\theta_2 = 40.63^{\circ}$
We can find the horizontal distance a light ray travels from the meter stick to the top of the water:
$\frac{x_2}{25~cm} = tan~^{\circ}$
$x_2 = (25~cm)(tan~40.63^{\circ})$
$x_2 = 21.4~cm$
We can find the mark on the meter stick:
$x = x_1+x_2$
$x = (43.3~cm)+(21.4~cm)$
$x = 64.7~cm$
(c) If the tank is full:
$\frac{x}{50~cm} = tan~30^{\circ}$
$x = (50~cm)(tan~40.63^{\circ})$
$x = 42.9~cm$
