What was Rome's reason for attacking Carthage in the Third Punic War?

The siege of Carthage was decided years before the Third Punic War (149–146 B.C.) actually took place. In the Roman Republic during this time period, there were Consuls and Censors that acted jointly as the supreme political authorities. Cato the elder who was Censor of Rome from 184–149 B.C., made a trip to Carthage in 157 B.C. to negotiate peace between Carthage and Numidia. When Cato returned to Rome, he decided to rally support for a Roman invasion of Carthage. He told the Roman Senate stories of how wealthy Carthage was, and that it was “crowded with vigorous young men, was abounded with immense quantities of gold and silver, and had prodigious magazines of arms and all war-like stories.” Cato then displayed to the Senate a bundle of African figs that were the biggest and most beautiful the Roman senators had ever seen. He then informed the Senate that only a three-day journey separated the Romans from the Carthaginians. Cato believed further that the Carthaginians were becoming too rich and too powerful and thus posed a continual threat to the prosperity of Rome. Cato often and famously told the Senate, “Ceterum censeo Carthaginem esse delendam” which translates to “However, I think that Carthage must be destroyed.”

Cato’s political opponent, Publius Corculum, a Consul, did not want Rome to go to war with the Carthaginians because he feared that his countrymen, who were already swelled with pride on account of their regional dominance, would lose all sense of decency and humanity should they begin conquering foreign nations. Furthermore, Corculum did not see the Carthaginians as an immediate threat to Roman prosperity overall. Corculum did, however, see them as dangerous enough to where he figured that an invasion into Carthaginian territory would prove disastrous for the Roman military, its government, and its people.

In the end, however, Cato the Elder’s arguments in favor of a Roman invasion of Carthage prevailed in the Senate. The official declaration of war that was proclaimed by the Senate asserted that Carthage broke a 50-year peace treaty that it had with Rome when it went to war with Numidia, a North African nation that was an ally of Rome. It should be noted that the Romans themselves instigated the battles between the Carthaginians and the Numidians so as to have a reason to invade Carthage. Furthermore, the Senate proclaimed that the Carthaginians mistreated the son of the king of Numidia in their battles with the Numidians, furthering the Romans’ justification for an invasion.
It was under these pretexts that the Romans invaded Carthage and the rest is history. By the end of 146 B.C., the Romans, having commanded an army of 80,000 soldiers, had completely annihilated Carthage and for the next 581 years Carthage remained subject to Roman rule.