Final answer:
To determine how long it takes $1600 to grow to $3200 at 5.25% interest compounded quarterly, we use the compound interest formula. After setting up the equation and using logarithms to solve for time, we find the number of years needed for the investment to double.
Step-by-step explanation:
Calculating Time to Double Investment with Compound Interest
To calculate how long it will take for an investment of $1600 to grow to $3200 with an interest rate of 5.25% compounded quarterly, we can use the formula for compound interest:
A = P(1 +\frac{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.
In this case, we want to find t when A is $3200, P is $1600, r is 0.0525, and n is 4 (since interest is compounded quarterly).
Here is the step-by-step calculation:
Set up the equation as 3200 = 1600(1 + 0.0525/4)^{4t}.
- Divide both sides by 1600 to get 2 = (1 + 0.0525/4)^{4t}.
- Use logarithms to solve for t:
t = \frac{\log(2)}{4 \cdot \log(1 + 0.0525/4)}
By plugging the values into a calculator, we find an approximate value for t. This calculation will give us the number of years it takes for the principal to double from $1600 to $3200.