Final answer:
To find the time it takes for $4562 to double with a 4.65% interest compounded monthly, we can use the compound interest formula and solve for the time variable.
Step-by-step explanation:
To determine how long it will take for an investment of $4562 to double with a monthly compounded interest rate of 4.65%, we use the formula for compound interest and solve for time (t).
The formula is A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial amount of money), r is the annual interest rate (decimal), n is the number of times that interest is compounded per year, and t is the time in years the money is invested for.
We know that to double the investment, A must be 2P. So, 2P = P(1 + r/n)^(nt). With P being $4562, r being 0.0465 (4.65% expressed as a decimal), and n being 12 (since the interest is compounded monthly), we can cancel out the Ps and solve for t using logarithms. This will allow us to find the number of years it takes for the investment to double.