Final answer:
To find the time required for $3000 to grow to $6561 with continuous compounding at an 8% interest rate, we use the continuous compounding formula and solve for t, resulting in approximately 12.78 years (d) t=12.78.
Step-by-step explanation:
To calculate the time t needed to save $6561 when $3000 is deposited, compounded continuously at an annual interest rate of 8%, we use the formula for continuous compounding:
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, in years.
- e is the base of the natural logarithm.
We are looking for t, so we rearrange the formula to solve for t:
t = ln(A/P) / r
Plugging in the values:
t = ln(6561/3000) / 0.08
Using a calculator to find the natural logarithm:
t ≈ ln(2.187) / 0.08 ≈ 12.78
Thus, the correct answer is d) t=12.78 years.