Final answer:
Using the Rayleigh criterion formula, the smallest resolvable distance between two speedboats by the radar system at a range of 7.2 km is approximately 54.65 meters.
Step-by-step explanation:
To determine the smallest distance two speedboats can be from each other and still be resolved as two separate objects by a radar system, we can use the formula for the radar's resolution, which depends on the wavelength of the radar signal and the diameter of the antenna.
The radar's resolution is based on the Rayleigh criterion, which can be represented by the formula Δθ = 1.22 λ / D, where Δθ is the minimum angular separation resolvable, λ is the wavelength, and D is the diameter of the antenna.
First, we convert the wavelength to meters: λ = 1.3 cm = 0.013 m. Then, we plug the values into the Rayleigh criterion formula to find the minimum angular separation:
Δθ = 1.22 × 0.013 m / 2.1 m ≈ 7.59 × 10^{-3} radians.
To find the smallest distance between two objects (speedboats) that the radar can resolve, we then use the relationship distance = range × Δθ. Given the range of the radar is 7.2 km, we convert this to meters (7200 m) and then calculate:
Distance = 7200 m × 7.59 × 10^{-3} ≈ 54.648 m.
Therefore, at a range of 7.2 km, the smallest distance at which two speedboats can be resolved as separate objects by this navy cruiser's radar system is approximately 54.65 meters.