Final answer:
To find the time required for $100,000 to grow to $213,022 at 4% interest compounded quarterly, apply the compound interest formula and solve for t. Rearrange and plug in the values to determine the number of years, rounding to the nearest whole year.
Step-by-step explanation:
The question asks to calculate the time it will take for an investment of $100,000 to grow to $213,022 with quarterly compounding at an annual interest rate of 4%. To solve this, we can use the compound interest formula: A = P(1 + \(\frac{r}{n}\))^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. t is the time the money is invested for, in years.
To find t, we rearrange the formula to solve for t: t = \(\frac{\log(\frac{A}{P})}{n \cdot \log(1 + \frac{r}{n})}\). Plugging in the values, we have: t = \(\frac{\log(\frac{213022}{100000})}{4 \cdot \log(1 + \frac{0.04}{4})}\). After calculating the above expression, we round the result to the nearest whole year to find that the investment will be worth $213,022 after t years.