Some distant star is to be observed by some telescope of diameter of objective lens \(a\), at an angular resolution of \(3.0\times 10^{-7}\) radian. If the wavelength of light from the star reaching the telescope is \(500\) nm, the minimum diameter of the objective lens of the telescope is ________ cm. (nearest integer)

The angular resolution \(\Delta \theta\) of a telescope is given by the formula:

\(\Delta \theta = \dfrac{1.22 \lambda}{a}\)

where \(\lambda\) is the wavelength of light and \(a\) is the diameter of the objective lens.

Given:
\(\Delta \theta = 3.0 \times 10^{-7}\) rad
\(\lambda = 500 \text{ nm} = 500 \times 10^{-9}\) m \(= 5 \times 10^{-7}\) m

Substituting the values into the formula:

\(3.0 \times 10^{-7} = \dfrac{1.22 \times 5 \times 10^{-7}}{a}\)

\(a = \dfrac{1.22 \times 5 \times 10^{-7}}{3.0 \times 10^{-7}}\)

\(a = \dfrac{6.1}{3.0}\) m

\(a = 2.0333...\) m

Converting the diameter into centimeters:

\(a = 2.0333... \times 100\) cm \(= 203.33\) cm

Rounding to the nearest integer, we get \(203\) cm.

Answer: \(203\)