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Calculate the time needed to save $6561 if $3000 is deposited today and compounded continuously at an annual interest rate of 8%.

a) t=9.78
b) t=10.78
c) t=11.78
d) t=12.78

1 Answer

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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.

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