To find the accumulated present value of an investment under a continuous money flow, we use the formula:

$PV = \int_0^T R e^{-rt} \, dt$

Where:

• $R = 7,000$ (continuous annual money flow),
• $r = 0.007$ (interest rate compounded continuously),
• $T = 9$ (time period in years).

Substitute the values into the formula:

$PV = \int_0^9 7000 e^{-0.007t} \, dt$

Step 1: Evaluate the integral

The integral of $e^{-rt}$ is: $\int e^{-rt} \, dt = \frac{e^{-rt}}{-r}$

Thus: $PV = 7000 \left[ \frac{e^{-0.007t}}{-0.007} \right]_0^9$

Step 2: Simplify the expression

Factor $-\frac{1}{0.007}$ outside: $PV = -\frac{7000}{0.007} \left[ e^{-0.007t} \right]_0^9$

$PV = -1000000 \left[ e^{-0.007t} \right]_0^9$

Step 3: Evaluate the bounds

$PV = -1000000 \left( e^{-0.007 \cdot 9} - e^{-0.007 \cdot 0} \right)$

$PV = -1000000 \left( e^{-0.063} - 1 \right)$

Using $e^{-0.063} \approx 0.9389$: $PV = -1000000 \left( 0.9389 - 1 \right)$

$PV = -1000000 \cdot (-0.0611)$

$PV = 61100$

Final Answer:

The accumulated present value of the investment is $61,100.

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Would you like me to explain further or explore alternative scenarios?
Here are 5 related questions for further exploration:

1. How does the result change if the interest rate increases to 1%?
2. What would the accumulated value be if the continuous flow was $10,000 annually?
3. How do you compute future value under similar continuous flow conditions?
4. How would the calculation change if the flow ceased after 5 years instead of 9 years?
5. Can this formula apply to investments with a varying flow rate? If so, how?

Tip: Always ensure the interest rate aligns with the compounding method when solving time value of money problems.