Question

If we double the diameter of a telescope's mirror, what happens to its light-gathering ability? View Available Hint(s) If we double the diameter of a telescope's mirror, what happens to its light-gathering ability?

Light-gathering ability doubles as well.
Light-gathering ability quadruples.
There is no change.
The light-gathering ability is cut in half.

Answer 1

Light-gathering ability quadruples.

Step-by-step explanation:

The light-gathering ability of a telescope depends on the area of the mirror, which is the surface presented to the light. Since these mirrors are circular (diameter is mentioned), we need to know what happens with the area of it when its diameter is doubled.

If r is the radius and d the diameter of a circle, its area will be:

`A=\pi r^2=\pi ((d)/(2) )^2=\pi((d^2)/(4))=(\pi d^2)/(4)`

We want to know now what happens if we double the diameter to 2d, so we calculate the new area A':

`A'=\frac {\pi (2d)^2}{4}=\frac {\pi 4d^2}{4}=\pi d^2`

Which is 4 times greater than the original area A, so we conclude that its light-gathering ability quadruples.