Problem
The formula $t=\frac{\ln m}{n \ln \left(1+\frac{r}{n}\right)}$ can be used to find the number of years $t$ required to multiply an investment $m$ times when $r$ is the per annum interest rate compounded $n$ times a year. Complete parts (a) through (c). (a) How many years will it take to double the value of an IRA that compounds annually at the rate of $8 \%$ ? \[ t=9.01 \] (Type an integer or decimal rounded to two decimal places as needed.) (b) How many years will it take to triple the value of a savings account that compounds daily at an annual rate of $10 \%$ ? \[ t=\square \] (Type an integer or decimal rounded to two decimal places as needed.)
Solution
Step 1 ：Given that the savings account compounds daily at an annual rate of 10%, we want to find out how many years it will take to triple the value of the savings account.
Step 2 ：Substitute the values m=3, r=0.10, and n=365 into the formula \(t=\frac{\ln m}{n \ln \left(1+\frac{r}{n}\right)}\).
Step 3 ：Calculate the result to get \(t = 10.98762776631521\).
Step 4 ：Round the result to two decimal places to get \(t = 10.99\).
Step 5 ：So, it will take approximately 10.99 years to triple the value of the savings account.
Step 6 ：Final Answer: \(\boxed{10.99}\)
