Final answer:
To find out how long it will take for Susan's $1,000 investment with 8% interest compounded quarterly to grow to $10,000, we used the compound interest formula. After calculating, we find it will take approximately 18.40 years, so the closest answer is 20 years.
Step-by-step explanation:
The student's question asks about the time it would take for an investment of $1,000 to grow to $10,000 with an annual interest rate of 8% compounded quarterly.
To solve this, we will 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 ($1,000).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time in years.
Plugging in the values, we have:
10,000 = 1,000(1 + 0.08/4)^(4t)
To find t, we need to rearrange and solve for t:
10 = (1 + 0.02)^(4t)
Take the natural logarithm of both sides:
ln(10) = ln((1 + 0.02)^(4t))
ln(10) = 4t * ln(1.02)
t = ln(10) / (4 * ln(1.02))
By calculating the above expression, we find that t is approximately 18.40 years.
Therefore, the closest answer to the options provided is (d) 20 years.