Final answer:
To determine the time for a $5000 investment to grow to $7100 with an annual interest rate of 8.2%, compounded monthly, the compound interest formula is used. After calculating, the time required is approximately 10.24 years.
Step-by-step explanation:
To find the time required for an investment of $5000 to grow to $7100 at an interest rate of 8.2 percent per year, compounded monthly, we can use the formula for compound interest. The formula to use is:
A = P(1 + rac{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.
Firstly, we convert the interest rate from a percentage to a decimal by dividing by 100, giving us 0.082. Since the interest is compounded monthly, n = 12. Now we can set up our equation:
$7100 = $5000(1 + rac{0.082}{12})^{12t}
We now need to solve for t:
rac{7100}{5000} = (1 + rac{0.082}{12})^{12t}
1.42 = (1 + 0.006833)^{12t}
To solve for t, we'll take the natural logarithm of both sides:
ln(1.42) = 12t * ln(1 + 0.006833)
t = rac{ln(1.42)}{12 * ln(1 + 0.006833)}
After calculating, we find that t is approximately
10.24 years.
Therefore, it will take approximately 10.24 years for the $5000 investment to grow to $7100 at an 8.2% annual interest rate, compounded monthly.