Final answer:
To calculate the time needed for Rodney's investment to grow to $500,000 at an annual compounded rate of 8.5%, we use the compound interest formula and solve for 't' using logarithms. Assuming annual compounding, the formula is rewritten and calculated, providing the number of years for the growth.
Step-by-step explanation:
To determine how long it will take for Rodney's investment portfolio to grow from $259,000 to $500,000 at an annually compounded rate of 8.5%, we will use the compound interest formula A = P(1 + r/n)^(nt). Here, 'A' is the amount of money accumulated after 'n' years, including interest. 'P' is the principal amount ($259,000), 'r' is the annual interest rate (0.085), 'n' is the number of times that interest is compounded per year, and 't' is the time in years.
Since the question doesn't specify how often the interest is compounded, we'll assume it's compounded once a year ('n' = 1). We'll then need to solve the compound interest formula for 't' to find out how many years it will take for the investment to grow to $500,000. To solve for 't', we'll take logarithms.
Starting with the formula:
$500,000 = $259,000 * (1 + 0.085/1)^(1*t)
We'll rearrange it and solve for 't' using logarithms:
t = log($500,000 / $259,000) / log(1 + 0.085)
Plugging the values into a calculator gives us the number of years needed for the investment to reach $500,000.