Final answer:
To calculate the time needed for $10 to become $100,000 at an annual rate of 7.25%, we correct and use the formula t = ln(A/P) / ln(1+r), resulting in approximately 131.53 years.
Step-by-step explanation:
To determine how long it will take for $10 to grow to $100,000 at an annual interest rate of seven and a quarter percent, we can use the given formula t = ln(A/P) / r, with the understanding that we need to correct the formula to the standard compound interest formula which is t = ln(A/P) / ln(1+r), since the original formula provided seems to have a typo.
Here A is the future value of the investment, P is the present value, and r is the annual interest rate (expressed as a decimal).
First, we convert the interest rate from a percentage to a decimal by dividing by 100: 7.25% = 0.0725.
Next, we plug the numbers into the corrected compound interest formula So we have: t = ln(100,000/10) / ln(1 + 0.0725) = ln(10,000) / ln(1.0725).
Using a calculator, we find that :Now we divide the natural log of the final amount by the natural log of 1 plus the rate: t = 9.21034 / 0.070024 ≈ 131.53 years.
It will take approximately 131.53 years for $10 to grow to $100,000 at an annual interest rate of seven and a quarter percent when compounded annually.