Question

Suppose astronomers built a 150-meter telescope. how much greater would its light-collecting area be than that of the 10-meter keck telescope?

Answer 1

Almost all telescopes have a circular mirror. The area of a circle is proportional to $r^2$ where $r$ is the radius of the circle, the constant of proportionality being $4\pi$
$S =4\pi r^2$
Therefore the area of 150 meter telescope would be 
$S_1/S_2=(r_1/r_2)^2 =(75/5)^2 =15^2=225$
times bigger than the area of the smaller (10 meter) telescope.