Joe deposits $1,500 in an account that pays 3% annual interest compounded continuously.How much will Joe have in his account after 5 years?How long will it take Joe to double his money? Use natural logarithms and explain your answer.
The formula for compound interest compounded continuously is given by the formula A = P*e^(rt), where A is the final amount, P is the principal amount (initial deposit), r is the annual interest rate (as a decimal), and t is the time in years.
In this case, Joe deposited $1,500, so P = 1500. The annual interest rate is 3%, so r = 0.03. The time is 5 years, so t = 5.
Using the formula, we have A = 1500*e^(0.03*5).
A simplification of the formula becomes: A = 1500*e^0.15.
Using a calculator, we find that e^0.15 is approximately 1.161828.
Therefore, Joe will have approximately A = 1500 * 1.161828 = $1,742.74 in his account after 5 years.
To calculate the time it will take for Joe to double his money, we need to solve the equation 2P = P*e^(rt) for t.
In this case, P = 1500 and A = 2P = 3000.
Our equation becomes: 3000 = 1500*e^(0.03t).
Dividing both sides by 1500: 2 = e^(0.03t).
Taking the natural logarithm (ln) of both sides: ln(2) = 0.03t.
Dividing both sides by 0.03: ln(2)/0.03 ≈ t.
Using a calculator, we find that ln(2)/0.03 ≈ 23.105.
Therefore, it will take Joe approximately 23.105 years to double his money.