Question

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

Answer 1

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:

$CollectingArea=\pi*(\frac{diameter}{2})^{2} =  \pi radius^{2}$

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:

$Keck=\pi *(\frac{10m}{2})^{2}=\pi *(5m)^{2}=25\pi m^{2}$

For the 5-meter hale telescope, we have:

$Keck=\pi *(\frac{5m}{2})^{2}=\pi *(2.5m)^{2}=6.25\pi m^{2}$

Now, comparing the areas, we have:

$Ratio=\frac{KecksArea}{HalesArea}=\frac{25\pi m^{2} }{6.25\pi m^{2}}=4$

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!