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The function f(x)=900 represents the rate of flow of money in dollars per year. Assume a 10 - year period at 5% compounded continuously. Find (A) the present value, and (B) the accumulated amount of money flow at t=10.

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Final answer:

The question involves calculating the present value and the accumulated amount of a continuous flow of money at a constant rate, given an annual interest rate compounded continuously over a specific period.

Step-by-step explanation:

The question pertains to the calculation of present value and future or accumulated amount of a constant money flow when compounded continuously at a given interest rate.

(A) To calculate the present value (PV) of a future sum of money, we can use the formula:


PV = F / e^(rt)

Where:

  • F is the future payment
  • e is the base of the natural logarithm (approximately 2.71828)
  • r is the annual interest rate (as a decimal)
  • t is the time in years

Since the flow of money is continuous, to find the present value of the money flow at t=10 years and r=5%, we would calculate the present value of each potential future payment of $900 and then sum them up over the 10 years.

(B) To find the accumulated amount at t=10 years, we use the formula for continuous compounding:

A = Pe^(rt)

Where:

  • A is the accumulated amount
  • P is the principal amount (in this case, the rate of flow of money = $900 per year)
  • r is the annual interest rate (as a decimal)
  • t is the time in years

So, the accumulated amount of money flow at t=10, with a rate of 5% compounded continuously, would be:

A = 900e^(0.05*10)

It is important to integrate the flow of money over the period to find the total value.

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