The correct answer is $\boxed{\text{A}}$.
The latitude of a place is the angle between the equatorial plane and the vertical at that place. The altitude of a star is the angle between the horizon and the star. The upper transit of a star is when the star crosses the observer’s meridian at its highest point in the sky. The lower transit of a star is when the star crosses the observer’s meridian at its lowest point in the sky.
The latitude of a place can be calculated using the following formula:
$$\lambda = \arctan \left(\frac{h_u – h_l}{2}\right)$$
where $\lambda$ is the latitude of the place, $h_u$ is the altitude of the star at its upper transit, and $h_l$ is the altitude of the star at its lower transit.
In this case, $h_u = 60^\circ 30’$ and $h_l = 19^\circ 30’$. Substituting these values into the formula gives:
$$\lambda = \arctan \left(\frac{60^\circ 30′ – 19^\circ 30′}{2}\right) = 30^\circ$$
Therefore, the latitude of the place is $\boxed{30^\circ}$.
Option B is incorrect because the altitude of the star at its upper transit is greater than the altitude of the star at its lower transit. This is not possible, as the star must be higher in the sky at its upper transit than it is at its lower transit.
Option C is incorrect because the altitude of the star at its upper transit is equal to the altitude of the star at its lower transit. This is only possible if the star is directly overhead, which is not the case in this situation.
Option D is incorrect because the altitude of the star at its upper transit is less than the altitude of the star at its lower transit. This is not possible, as the star must be higher in the sky at its upper transit than it is at its lower transit.