The correct answer is:
94.8377 years
Explanation:
The formula for compound interest is
\(A=p(1+\frac{r}{n})^{nt}\), where A is the total amount, p is the amount of principal deposited, r is the interest rate as a decimal number, n is the number of times per year the interest is compounded, and t is the number of years.
For our problem, A = 3000; p = 175; r = 3% = 3/100 = 0.03; n = 12; and t is unknown:
\(3000=175(1+\frac{0.03}{12})^{12t}
\\3000=175(1+0.0025)^{12t}
\\3000=175(1.0025)^{12t}\)
Divide both sides by 175:
\(\frac{3000}{175}=\frac{175(1.0025)^{12t}}{175}
\\
\\\frac{120}{7}=1.0025^{12t}\)
We will use logarithms to solve this. The base of the exponent is 1.0025, so this will be the base of the log:
\(\log_{1.0025}(\frac{120}{7})=12t\)
Divide both sides by 12:
\(\frac{\log_{1.0025}(\frac{120}{7})}{12}=\frac{12t}{12}
\\
\\94.8377\)
