Following the birth of a​ child, a parent wants to make an initial investment Upper P 0 that will grow to ​$50 comma 000 for the​ child's education at age 19. Interest is compounded continuously at 6​%.
What should the initial investment​ be? Such an amount is called the present value of ​$50 comma 000 due 19 years from now.
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Part 1
The present value is about ​$
  
enter your response here.
​(Do not round until the final answer. Then round to two decimal places as​ needed.)

To find the present value $ P_0 $ of an amount $ A $ due in the future, compounded continuously at a rate $ r $, you can use the formula:

\[
P_0 = A e^{-rt}
\]

Where:
- $ A = 50,000 $ (the future amount)
- $ r = 0.06 $ (the interest rate)
- $ t = 19 $ (the number of years)

Now, substituting in the values:

\[
P_0 = 50000 \times e^{-0.06 \times 19}
\]

First, calculate the exponent:

\[
-0.06 \times 19 = -1.14
\]

Now calculate $ e^{-1.14} $:

\[
e^{-1.14} \approx 0.3208
\]

Now substitute back into the formula:

\[
P_0 = 50000 \times 0.3208 \approx 16040
\]

Finally, rounding to two decimal places, the present value is approximately:

\[
P_0 \approx 16040.48
\]

So, the initial investment should be about **$16,040.48**.

Would you like further details on this calculation or have any questions? Here are some related questions you might consider:

1. How does the present value change with different interest rates?
2. What is the effect of increasing the investment period on the present value?
3. How would this calculation differ if the interest were compounded annually instead of continuously?
4. What is the future value if you invested a different amount with the same interest rate?
5. How can this concept be applied to other financial goals besides education? 

**Tip:** Always ensure you understand the difference between present value and future value when planning investments.

Following the birth of a child, a parent wants to make an initial investment Upper P 0 that will grow to $50,000 for the child's education at age 19. Interest is compounded continuously at 6%. What should the initial investment be?

Discover how to calculate the present value needed for an investment that will grow to $50,000 for a child's education at age 19, with continuous compounding at a 6% interest rate. This detailed solution explains the formula and calculation steps.

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