Introduction:
In the given question, the father rants about his agony to his son. We need to determine the age of the son when this event occurs.

Explanation:
To determine the age of the son when his father rants about his agony, we can analyze the options provided:

a) Teenager:
A teenager is typically between the ages of 13 and 19. During this period, children experience significant physical, emotional, and cognitive development. They may have a better understanding of their parents' emotions and can engage in meaningful conversations. However, the term "teenager" is quite broad, and it is unclear if the son is at the beginning or end of this age range. Therefore, this option is not definitive.

b) Grown-up:
A grown-up generally refers to an individual who has reached adulthood, usually beyond the teenage years. They have likely completed their education and have more life experience. Being a grown-up, the son would have a mature perspective and be able to engage in deep conversations about his father's agony. Therefore, this option seems plausible.

c) Child:
A child is typically younger than a teenager, ranging from infancy to early adolescence. While children can show empathy and understanding, they may not have the same level of emotional maturity as a grown-up. Therefore, it is less likely that the father would rant about his agony to such a young child.

d) Just became adult:
If the son has just become an adult, he would be at the beginning of his adult life, around the age of 18 or 19. While he may have a level of maturity and understanding, he may still be transitioning into adulthood and may not have fully developed the capacity for deep conversations. Therefore, this option is less likely.

Conclusion:
Considering the options given, it is most likely that the son is a grown-up when his father rants about his agony. A grown-up would have the necessary emotional maturity and life experience to engage in meaningful discussions about his father's troubles.