Final Answer:
No, the eye cannot resolve the two towers with the given parameters.
Step-by-step explanation:
The ability of an optical system, such as the human eye or a telescope, to distinguish between two close objects is determined by the Rayleigh criterion. According to Rayleigh's criterion, two point sources are just resolvable when the central maximum of the diffraction pattern of one source coincides with the first minimum of the diffraction pattern of the other source.
The angular resolution (θ) can be calculated using the formula θ = 1.22 * (λ/D), where λ is the wavelength of light and D is the diameter of the aperture (pupil in the case of the eye). In this scenario, using a wavelength of 550 nm and a pupil diameter of 4.0 mm, the angular resolution is calculated to be approximately 1.40 x 10^(-4) radians.
Now, to determine if the two towers are resolvable, we need to find the angular separation between them. The angular separation (δθ) can be calculated using the formula δθ = s/D, where s is the distance between the towers. With a distance of 20.0 m, and considering the city is 12 km away, the angular separation is approximately 8.33 x 10^(-4) radians.
Since the angular separation is greater than the angular resolution, the eye cannot resolve the two towers. To find the minimum magnification power needed to resolve them, we can use the formula M = 1/θ, where M is the magnification. Substituting the values, we find that the minimum magnification required is approximately 7.14x. Therefore, a telescope with a magnification power greater than 7.14x would be needed to resolve the two towers under the given conditions.