Astronomers are proud of their telescopes and often say they can "resolve a dime at so many miles." How many miles from a $\text { 102-cm-diameter }$ telescope mirror can you place a coin of diameter 3.0 cm and just resolve the coin using light of wavelength $389 \mathrm {~nm} ?$ (1 mi = 1.609 km)
A) 19 mi
B) 9.7 mi
C) 40 mi
D) 190,000,000 mi

Question

Quizplus · Accepted Answer

The resolving power of a telescope is given by the formula $\theta = 1.22 \frac{\lambda}{D}$, where $\lambda$ is the wavelength and $D$ is the diameter of the telescope. Converting the diameter of the telescope to meters and the wavelength to meters, we can calculate the angular resolution. Then, using the small angle approximation, the distance can be calculated by $d = \frac{r}{\theta}$, where $r$ is the diameter of the coin. Converting the result to miles gives approximately 19 miles.

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