Final answer:
An investment of $2900 at an 8% interest rate compounded monthly will take approximately 8.7 years to double. This is calculated using the compound interest formula and solving for the time variable t using natural logarithms.
Step-by-step explanation:
To find out how long it takes for a $2900 investment to double at an 8% interest rate compounded monthly, we use the formula for compound interest A = P(1 + 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.
Since we want to double our principal, A = 2P.
Thus, our equation would be 2P = P(1 + r/n)^(nt), which simplifies to 2 = (1 + r/n)^(nt).
We can use logarithms to solve for t.
Given: P = $2900, r = 0.08 (8%), n = 12 (monthly compounding)
We need to solve the equation 2 = (1 + 0.08/12)^(12t).
Taking the natural logarithm of both sides gives us ln(2) = 12t * ln(1 + 0.08/12).
Therefore, t = ln(2) / (12 * ln(1 + 0.08/12)).
Calculating this gives us a t value of approximately 8.661, which rounded to the nearest tenth is 8.7 years. So, the correct answer is 8.7 years.