Final answer:
This calculation involves continuous compound interest. The $(A)t$ is calculated using the formula $A(t) = Pe^{rt}$. The differential equation representing this situation is dA/dt = rA(t), and by plugging in various values into the equations, you can know the balance in the account after certain years, when it will reach a specific amount, and how fast it's growing at a certain point.
Step-by-step explanation:
The question is about understanding the principle of continuous compounding in finance, here, with an initial deposit of five thousand dollars at an annual interest rate of 2.5% compounded continuously.