Final answer:
To find out how long it takes for $5,000 to grow to $10,200 with a 5.25% interest rate compounded semi-annually, we can use the formula A = P(1 + r/n)^(nt). Solving this equation, we find that the time required is approximately 8.3 years.
Step-by-step explanation:
To find out how long it takes for the money to reach $10,200, we need to determine the time required to grow from $5,000 to $10,200 with a 5.25% interest rate compounded semi-annually.
We can use the formula:
A = P(1 + r/n)^(nt)
where:
- A = future value
- P = principal (initial investment)
- r = annual interest rate (as a decimal)
- n = number of compounding periods per year
- t = number of years
In this case, P = $5,000, A = $10,200, r = 5.25% = 0.0525, and n = 2 (semi-annual compounding).
Substituting these values into the formula, we get:
$10,200 = $5,000(1 + 0.0525/2)^(2t)
Dividing both sides by $5,000, we get:
2.04 = (1.02625)^(2t)
Taking the logarithm of both sides, we get:
log(2.04) = log(1.02625)^(2t)
Using the logarithm properties, we can simplify:
2t = log(2.04) / log(1.02625)
Finally, solving for t, we get:
t = (log(2.04) / log(1.02625)) / 2
Calculating the right-hand side of the equation gives us approximately 8.3. Therefore, the person must leave the money in the bank for 8.3 years (to the nearest tenth of a year) to reach $10,200.