A woman cannot see objects closer than 40 cm from her left eye and...

The woman is suffering from hypermetropia, so she uses convex lenses for reading.

For the left eye, the near point is $40 \text{ cm}$. Using the lens formula with $v = -40 \text{ cm}$ and $u = -25 \text{ cm}$:

$\dfrac{1}{f_{L}} = \dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{-40} - \dfrac{1}{-25} = \dfrac{3}{200} \text{ cm}^{-1}$

$f_{L} = \dfrac{200}{3} \text{ cm}$

For the right eye, the near point is $100 \text{ cm}$. Using the lens formula with $v = -100 \text{ cm}$ and $u = -25 \text{ cm}$:

$\dfrac{1}{f_{R}} = \dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{-100} - \dfrac{1}{-25} = \dfrac{3}{100} \text{ cm}^{-1}$

$f_{R} = \dfrac{100}{3} \text{ cm}$

Since $f_{L} > f_{R}$, the left lens has a larger focal length than the right lens.

In an astronomical telescope, the magnifying power is given by $M = \dfrac{f_{o}}{f_{e}}$. To achieve high magnification, the objective lens should have a larger focal length and the eyepiece should have a smaller focal length.

Therefore, the right lens, which has the smaller focal length, should be used as the eyepiece.