An investor deposits 10,000 dollars into an account that earns interest at a constant continuous rate. No further deposits or withdrawals are made.
(a) Assuming the account grows according to the differential equation dtdA=0.05A, where A(t) is the account balance in dollars and t is measured in years, determine the time required for the balance to reach 18,000 dollars. Express your answer to the nearest hundredth of a year.
(b) Describe what would happen to the time found in part (a) if the interest rate were increased to 6% while all other conditions remained the same. Justify your conclusion without performing additional numeric calculations.