From part d, you know that the galaxy's rotation curve is nearly flat, even well beyond the point at which we see any stars in the galaxy. according to the law of gravity as we understand it today, the only explanation for these this flatness is that

substantial amounts of mass must reside at great distances from the galactic center

Explanation:

In the galaxy rotation curve the plot is made by comparing the orbital speed of the luminous matter and the radial velocity of stars. According to the theoretical calculations i.e., when the visible mass of the galaxy is used to plot the curve the curve appears to increase then decrease. When the observable rotational speed of the visible matter is considered the curve increases then flattens.

This difference in the curves can be attributed to the existence of dark matter which is a substantial amount. The dark matter is generally situated in the dark matter halo which is at a great distance from the galactic center.

answer: at 24 february, 2014 the population gets twice of its initial population.

explanation:

since we have given that

the population for alpha city, t years after january 1, 2004 is given as

first we find out the initial population, i.e. t=0,

so, our quadratic equation becomes,

according to question, we have said that if the population has twice its initial population, then it becomes

so, time taken to reach this above mentioned population is given by

now, we know that time can't be negative so, we take t=9.149 years.

hence, at 24 february, 2014 the population gets twice of its initial population.

the rearrangement of particles in a physical change. hope this