Why doesn't a person cease to exist as soon as becoming an arahant? From what I know body and mind are two separate entities. Body cannot survive without a mind but is not the same case for a mind from what I know. When a person completely rids himself/herself of all the delusion and desire he/she becomes an arahant. An arahant ceases to exist when he/she passes away and nothing goes on or lingers and there are no more births. I am curious to know why this cannot happen as soon as becoming an arahant. Why does an arahant only cease to exist after death? How come the mind or sansara is extended to the point of which the body dies? Why doesn't the mind cease to exist as soon as attaining nirvana?
The mind cannot exist without a body except in the Arupa Brahma realms. 
Arahanths have no desire to die. So becoming enlightened does not imply that they have to immediately die. It's like when you are driving a car, if it runs out of fuel, it's not going to stop immediately. The momentum will take it some distance before stopping.