$4-$ An economics professor wants to hire a research assistant. She wishes to separate applicants into two groups $-$ those who are truly interested in research, and those who are only interested because the payment is good $($$$1,000).$ Let's call these two groups "interested" and "nonchalant," respectively. In an attempt to separate the two groups, she devises a plan under which potential RAs must slay dragons before applying for the job.

Those who slay the requisite number of dragons, $D$ will be allowed to apply. To a member of either group $($interested and nonchalant$),$ the benefit is equal to $$1,000.$ Those who do not slay the requisite number of dragons will get $$64$ for their effort.

To a member of the "interested" group, who pursue their goal with unbridled passion, the cost of passing test is given as $D^2,$ where $D$ is the number of dragons slain.

To a member of the "nonchalant" group, who pursue their goal halfheartedly, the cost of passing test is given as $D^{\text{'}},$ where $D$ is the number of dragons slain.

a$)$ Professor wants to sort the interested students from the nonchalant students. What is the minimum number of dragons professor can ask potential RAs to slay if she wants them to separate into groups? You can round to an appropriate integer.

b$)$ Suppose that professor asks students to slay three fewer dragons than you indicated in your answer to $($a$).$ Why will asking students to slay this many dragons not help professor filter out the nonchalant students?

c$)$ What is the maximum number of dragons professor can ask potential RAs to slay if she wants to be able to distinguish between interested and nonchalant students? $($Again$,$ you may round your answer to an appropriate integer.$)$

d$)$ Suppose that professor asks potential RAs to slay three more dragons than you indicated in your answer to $($c$).$ Why will asking students to slay this many dragons not help professor filter out the nonchalant students?