Determine the position of the image produced by refraction in the...

From the given figure and sign convention, the parameters for refraction at the spherical surface are:
Refractive index of the first medium, $\mu_1 = 1.0$
Refractive index of the second medium, $\mu_2 = 1.5$
Object distance, $u = -25 \text{ cm}$
Radius of curvature, $R = +20 \text{ cm}$

The formula for refraction at a single spherical surface is:
$\dfrac{\mu_2}{v} - \dfrac{\mu_1}{u} = \dfrac{\mu_2 - \mu_1}{R}$

Substituting the given values:
$\dfrac{1.5}{v} - \dfrac{1.0}{-25} = \dfrac{1.5 - 1.0}{+20}$

$\dfrac{1.5}{v} + \dfrac{1}{25} = \dfrac{0.5}{20}$

$\dfrac{1.5}{v} + \dfrac{1}{25} = \dfrac{1}{40}$

Rearranging the terms to solve for $v$:
$\dfrac{1.5}{v} = \dfrac{1}{40} - \dfrac{1}{25}$

$\dfrac{1.5}{v} = \dfrac{5 - 8}{200}$

$\dfrac{1.5}{v} = \dfrac{-3}{200}$

$v = \dfrac{1.5 \times 200}{-3}$

$v = -100 \text{ cm}$

The negative sign indicates that the image is virtual and is formed on the same side as the object. Thus, the image is located $100 \text{ cm}$ from the surface on the side of $S$.