Final answer:
An investment of $1000 at a 5.25% continuous compounding rate will take approximately 7.7 years to grow to $1500, corresponding to option C in the question.
Step-by-step explanation:
The student asked when an investment of $1000 would be worth $1500 if the balance is compounded continuously at a rate of 5.25%. This is related to the concept of exponential growth and can be solved using the continuous compounding formula, which is A = Pert, 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), t is the time the money is invested for, and e is the base of the natural logarithm.
To solve for the time (t), we rearrange the formula to solve for t:
t = (ln(A/P)) / r
In this case:
A = $1500
P = $1000
r = 5.25% or 0.0525
t = (ln(1500/1000)) / 0.0525
t = (ln(1.5)) / 0.0525
t ≈ 7.7 years
Therefore, the investment will be worth $1500 after approximately 7.7 years, which corresponds to option C.