Why were epicycles necessary in ptolemy’s model of the universe?

Because Mars, Jupiter, and Saturn each appear to slow down, stop, move backwards for a while, then again slow down, stop, and start moving forward again. Each plant does this at some time during each year.

If they were simply moving in circles around the Earth, they couldn't do any of this. They would need to be moving on small circles attached to the big circles. The small ones are the 'epicycles'.

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.