Viewed from Earth, the Moon subtends an angle of \(0.5^{\circ}\) in the sky. How large an image of the Moon will be formed by the \(3.6-\mathrm{m}-\) diameter, 8.5 -m-focal-length mirror of the Canada-FranceHawaii Telescope?

Angular size is a measurement of the angle formed by lines extending from the observer to the edges of an object. It's how large an object appears to us based on its actual size and distance from us. In astronomy, this concept is crucial for determining how big celestial bodies, like stars and planets, appear when viewed from Earth. Imagine pointing at the object with your arms outstretched; the angular size is the angle between your arms.

For example, when looking at the moon, even though it is much smaller than the sun, it can appear similar in size because it is closer to Earth, which increases its angular size. This concept is at the heart of the exercise, where the moon's angular size allows us to calculate the image size produced by a telescope.

The subtended angle is the angle that a viewed object occupies at the observer's eye, connecting two lines from the eye to the extremities of the object. You can think of it as 'how much space' the object takes up in your field of vision.

Often expressed in degrees, the subtended angle is the starting point for calculating the image size in our telescope example. In the exercise, the moon's subtended angle is given as 0.5 degrees, which is a measure of how broad the moon looks to us from Earth.

Focal length is a fundamental concept in optics, representing the distance over which parallel rays of light converge after passing through a lens or reflecting off a mirror. The focal length of a telescope determines how far away from the lens or mirror the image will focus.

In the context of the exercise, the Canada-France-Hawaii Telescope has a focal length of 8.5 meters, meaning that parallel rays of light coming from the moon (essentially at an infinite distance) will form an image of the moon 8.5 meters behind the telescope's mirror. The focal length directly influences the size of the image formed by magnifying the subtended angle.

Radians are an alternative unit to degrees for measuring angles, central to various fields of math and science. They offer a direct connection to the arc length on a circle's circumference, making calculations involving circular motion and angles more natural and straightforward.

To convert the subtended angle from degrees to radians, as required in the exercise, we use the conversion factor \( \frac{\pi} {180} \) radians per degree, because there are \( 2\pi \) radians in a full circle, equivalent to 360 degrees. Performing this calculation for the 0.5-degree angle yields a result in radians, which can then be applied to the formula to find the image size.