Refraction at a curved surface: A fish appears to be 2.00 m below the surface of a pond when viewed almost directly above by a fisherman. What is the actual depth of the fish? (n_water = 1.33)

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The actual depth of the fish can be found using the formula for refraction at a curved surface, which is $n_1/v + n_2/u = (n_2 - n_1)/R$, where $n_1$ and $n_2$ are the refractive indices of the medium (air and water, respectively), $v$ is the image distance (apparent depth), and $u$ is the object distance (actual depth). However, for a flat surface, this simplifies to Snell's law, which can be expressed as $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$. For viewing almost directly above, $\theta_1$ and $\theta_2$ are small, and the sine of the angles can be approximated by the tangent of the angles, leading to the simpler relation $n_1/v = n_2/u$. Rearranging for $u$, we get $u = n_2 \cdot v / n_1$. Substituting the given values ($n_1 = 1$ for air, $n_2 = 1.33$ for water, and $v = 2.00$ m), we find $u = 1.33 \cdot 2.00 / 1 = 2.66$ m.

Refraction at a Curved Surface: a Fish Appears to Be