Introduction:
The question asks about the ratio of maximum stress induced in a thin cylindrical pressure vessel and a thin spherical pressure vessel. Both vessels have the same internal diameter, wall thickness, material, and internal pressure. We need to determine the relationship between the maximum stress in these two types of vessels.

Explanation:
To understand the relationship between the maximum stress in a thin cylindrical pressure vessel and a thin spherical pressure vessel, we can analyze the stress distribution in each type of vessel.

Stress Distribution in a Thin Cylindrical Pressure Vessel:
In a thin cylindrical pressure vessel, the stress distribution is uniform across the wall thickness. The maximum stress in a thin cylindrical pressure vessel occurs on the inner surface of the cylinder. This maximum stress can be calculated using the formula:

Maximum Stress in a Thin Cylindrical Pressure Vessel:
σ_cylindrical = P * r / t

Where:
σ_cylindrical = Maximum stress in the thin cylindrical pressure vessel
P = Internal pressure
r = Internal radius of the vessel
t = Wall thickness of the vessel

Stress Distribution in a Thin Spherical Pressure Vessel:
In a thin spherical pressure vessel, the stress distribution is not uniform across the wall thickness. The stress is highest at the equator and decreases towards the poles. The maximum stress in a thin spherical pressure vessel occurs at the equator. This maximum stress can be calculated using the formula:

Maximum Stress in a Thin Spherical Pressure Vessel:
σ_spherical = P * r / (2t)

Where:
σ_spherical = Maximum stress in the thin spherical pressure vessel
P = Internal pressure
r = Internal radius of the vessel
t = Wall thickness of the vessel

Comparison between Maximum Stresses:
To compare the maximum stresses in the two types of vessels, we can calculate the ratio:

Ratio of Maximum Stresses:
σ_cylindrical / σ_spherical = (P * r / t) / (P * r / (2t))
= (2t * P * r) / (t * P * r)
= 2

Therefore, the ratio of the maximum stress induced in a thin cylindrical pressure vessel to that in a thin spherical pressure vessel is 2. In other words, the maximum stress in a thin cylindrical pressure vessel is twice that in a thin spherical pressure vessel for the same internal diameter, wall thickness, material, and internal pressure.

Conclusion:
The correct answer is option 'A' - the ratio of maximum stress induced in a thin cylindrical and thin spherical pressure vessel is 2.