To resolve the image of Earth as an exoplanet, we need a telescope with a minimum diameter calculated using the formula D = 1.22 * λ / α, where λ is the wavelength of light and α is the angular resolution required. The formula α = size / distance can be used to calculate the angular resolution needed to resolve details. In this case, the distance is 384,000 km and the size is 5.00 km.
Step-by-step explanation:
In order to resolve the image of the Earth as an exoplanet, we need a telescope with a diameter large enough to achieve a resolution that can differentiate details as small as 5.00 km. The diameter of the telescope can be calculated using the formula:
D = 1.22 * λ / α
Where D is the diameter of the telescope, λ is the wavelength of light (given as 500 nm in this case), and α is the angular resolution needed to resolve details (in radians).
Since the angular resolution can be calculated as the size of the object divided by the distance to the object, we can use the formula:
α = size / distance
In this case, the distance is 384,000 km and the size is 5.00 km. Substituting the values into the formulas, we can calculate the minimum diameter of the telescope required to resolve the image.