This exercise is based on the following table, which lists interest rates on long-term investments (based on 10-year government bonds) in several countries in 2008. HINT [See Example 4.] (Round your answer to the nearest integer.)
If you invest $10,200 in the United States, and the interest is compounded monthly, how many months will it take for your investment to grow to $20,000?
months
Country U.S. Japan Canada Germany Australia
Yield 3.9% 1.5% 3.8% 4.3% 5.9%
This sounds like a finance or math question, not an sfsu question.
12
20
To solve this problem, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment ($20,000 in this case)
P = the principal amount ($10,200 in this case)
r = the interest rate (3.9% in this case)
n = the number of times the interest is compounded per year (monthly compounding in this case)
t = the number of years
Since we are looking for the number of months, we can convert it to years by dividing the total number of months by 12.
Let's plug in the values:
20000 = 10200(1 + 0.039/12)^(12t)
Now, we can solve for t using logarithms:
First, divide both sides by 10200:
20000/10200 = (1 + 0.039/12)^(12t)
Simplifying:
2 = (1 + 0.039/12)^(12t)
Take the logarithm of both sides:
log(2) = log((1 + 0.039/12)^(12t))
Using the logarithmic property, we can bring the exponent down:
log(2) = 12t * log(1 + 0.039/12)
Now, divide both sides by 12 * log(1 + 0.039/12):
log(2) / (12 * log(1 + 0.039/12)) = t
Calculating this expression will give us the value of t (in years). As mentioned earlier, we need the answer in months, so we will multiply the result by 12.
Therefore, to find the number of months it will take for the investment to grow to $20,000, calculate:
t = log(2) / (12 * log(1 + 0.039/12))
months = t * 12
Round your answer to the nearest integer for the final result.