Final answer:
To find the required interest rate for an investment to grow from $4,000 to $6,500 in 9 years, use the compound interest formula with different values of 'n' depending on the compounding period (annually n=1, quarterly n=4) and solve for the interest rate 'r'.
Step-by-step explanation:
To find the interest rate required for an investment of $4,000 to grow to $6,500 in 9 years, whether compounded annually or quarterly, we use the compound interest formula. The compound interest formula is A = P(1 + r/n)^(nt), where:
- A is the future value of the investment.
- P is the principal amount, which is $4,000.
- r is the annual interest rate (expressed as a decimal), which we need to find.
- n is the number of times the interest is compounded per year.
- t is the time in years, which is 9 years.
For part a, where interest is compounded annually, n is 1.
For part b, where interest is compounded quarterly, n is 4.
Now, let's solve for r in both scenarios:
Annually Compounded Interest
After plugging in the values into the formula, we have $6,500 = $4,000(1 + r/1)^(1*9). Simplifying the equation, we solve for r and get the required annual interest rate.
Quarterly Compounded Interest
Here, the calculation changes slightly with $6,500 = $4,000(1 + r/4)^(4*9). After simplifying and solving for r, we find the required interest rate for quarterly compounding.