How much greater is the light-collecting area of one of the 10-meter keck telescopes than that of the 5-meter hale telescope?

The light-collecting area of the 10-meter Keck telescope is 4 times greater than the light-collecting area of the 5-meter Hale telescope.

Why?

We can calculate the light-collecting area of a telescope by using its diameter/radius.  To do that, we can use the following formula:

Now, to know how much greater is the collecting area of the 10-meter keck telescope compared to the collecting area of the 5-meter hale telescope, we need to calculate their light-collecting areas and compare them.

For the 10-meter keck telescope, we have:

For the 5-meter hale telescope, we have:

Now, comparing the areas, we have:

Hence, we have that the light-collecting area of the 10-meter keck telescope is 4 times greater than the light-collecting area of the 5-meter hale telescope.

Have a nice day!

the working equation for the angular size is the product of the physical size and 360 degrees divided by the product of two pi and distance from earth. the minimum size would be 0.559 degrees while the maximum size would be 0.49 degrees.