1. (3) Scaling is a pretty straightforward tool that only uses simple multiplication and division. Sometimes the scaling is linear, so an example is if a single can of soda holds 12 oz, how many does a six pack hold (6x12 oz = 72 oz). Another example is volume, which scales with the size cubed. If you have a 12 oz can of soda, you can ask how much more soda can you fit into a bottle that was twice as large (in each dimension). The volume increases as size cubed, so the volume is larger by 2x2x2 = 8. You can fit 96 oz of soda (8 times as much) in a container that is only twice the size of your can. Suppose that all the stars around us were the same intrinsic brightness (same luminosity) and they have a uniform density in space (number of stars per cubic pc), so that the further we can see, the more stars become visible. With our eyes, we see about 6000 stars if we go to a really dark site, and these stars are all within a distance of 25 parsecs (pc). We wish to study one million stars in order to get a good idea of the entire range of stars that exist.

A. Out to what distance must we observe in order to get that many? You can break this part into pieces and first begin by asking whether we expect the new distance to be smaller or larger than the 25 pc distance that is the limit with our eyes. Think in very simple terms about how the number of visible stars relates to the volume. Then, how does the volume relate to the distance? You can either try a variety of distances until you get a result close to the goal of 106 stars, or you can solve for the result (either way is fine).