Final answer:
To determine the time to double a $5000 deposit at a rate of 9.25% compounded daily, we use the compound interest formula and solve for time (t), which is approximately 7.49 years, making option B) the correct answer.
Step-by-step explanation:
To determine how long it will take for a $5000 deposit to double in value with an interest rate of 9.25% compounded daily, we can use the formula for compound interest. The formula is A = P(1 + r/n)^(nt), where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial sum 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 in years.
To double the original deposit of $5000, we set A to $10000 and solve for t. The annual interest rate of 9.25% as a decimal is 0.0925, and since the interest is compounded daily, n is 365. Plugging the values into the formula gives us:
10000 = 5000(1 + 0.0925/365)^(365t)
To solve for t, we first divide both sides by 5000:
2 = (1 + 0.0925/365)^(365t)
Now, take the natural logarithm of both sides:
ln(2) = 365t * ln(1 + 0.0925/365)
And solve for t:
t = ln(2) / (365 * ln(1 + 0.0925/365))
Calculate the right side to find t:
t ≈ 7.49 years
Therefore, the correct answer is B) Approximately 7.49 years.