A ray of light makes a glancing angle of incidence of 60° with a flat glass surface. Given that the critical angle for glass is 42°, determine the angle of refraction r.

$$n_g = \frac{c}{v_g}$$

Substitute values:

$$n_g = \frac{3 \times 10^{8}}{2 \times 10^{8}} = 1.5$$

$$n_{\text{air}} \sin i = n_g \sin r$$

Substitute $n_{\text{air}} = 1$, $i = 45^\circ$:

$$1 \cdot \sin 45^\circ = 1.5 \sin r$$ $$\sin r = \frac{\sin 45^\circ}{1.5}$$

$$\sin r = \frac{\frac{\sqrt{2}}{2}}{1.5} = \frac{\sqrt{2}}{2} \times \frac{2}{3} = \frac{\sqrt{2}}{3}$$ $$\sin r = \frac{1.414}{3} = 0.4713$$

$$r = \sin^{-1}(0.4713) \approx 28^\circ$$

Final answer: $r = 28^\circ$

$$L = v_o + f_e$$

Substitute values:

$$16.0 = v_o + 2.5$$

$v_o = 16.0 - 2.5 = 13.5$ cm

$$\frac{1}{v_o} - \frac{1}{u_o} = \frac{1}{f_o}$$ $$\frac{1}{u_o} = \frac{1}{f_o} - \frac{1}{v_o}$$

Substitute values:

$$\frac{1}{u_o} = \frac{1}{0.50} - \frac{1}{13.5} = 2 - \frac{2}{27}$$ $$\frac{1}{u_o} = \frac{54}{27} - \frac{2}{27} = \frac{52}{27}$$

$u_o = \frac{27}{52}$ cm

$$m_o = \frac{v_o}{|u_o|}$$

Substitute values:

$$m_o = \frac{13.5}{\frac{27}{52}} = 13.5 \times \frac{52}{27}$$ $$13.5 = \frac{27}{2}$$ $$m_o = \frac{27}{2} \times \frac{52}{27} = \frac{52}{2} = 26$$

$$m_e = \frac{D}{f_e}$$

Substitute values:

$$m_e = \frac{25}{2.5} = 10$$

$$m = m_o \times m_e$$

Substitute values:

$$m = 26 \times 10 = 260$$

Final answer: $m = 260$