Explanation:
To find out how long it will take for $4,000 to grow to $28,200 at an annual interest rate of 7.25%, you can use the formula for compound interest:
\[A = P(1 + r/n)^(nt)\]
Where:
A = the future value of the investment ($28,200 in this case)
P = the principal amount ($4,000)
r = the annual interest rate (7.25% or 0.0725 as a decimal)
n = the number of times the interest is compounded per year (assuming it's compounded annually, so n = 1)
t = the number of years
Now, plug in the values and solve for t:
\[28,200 = 4,000(1 + 0.0725/1)^(1*t)\]
Simplify:
\[28,200 = 4,000(1.0725)^t\]
Now, you need to solve for t. Divide both sides by $4,000:
\[28,200/4,000 = (1.0725)^t\]
\[7.05 = 1.0725^t\]
To solve for t, you can take the natural logarithm (ln) of both sides:
\[ln(7.05) = ln(1.0725^t)\]
Use the property of logarithms that allows you to bring down the exponent:
\[ln(7.05) = t * ln(1.0725)\]
Now, solve for t:
\[t = ln(7.05) / ln(1.0725) ≈ 31.23\]
Rounded to the nearest year, it will take approximately 31 years for the account value to reach $28,200.