Boris and Bela invest at 16% per annum compounded continuously. They make an original investment. After 5 years they withdraw an amount of money, which is 20% of the original investment amount. 13 years after the original investment their account balance is $70,000. How much was the withdrawal? Your final answer should be correct to 2 places after the decimal point. wHAT IS amount of the withdrawal is?

We are given a continuously compounded interest problem, where the investment is compounded at 16% per annum. We will use the formula for continuous compounding:

\[
A = P e^{rt}
\]

Where:
- $ A $ is the amount after time $ t $,
- $ P $ is the initial investment (principal),
- $ r $ is the annual interest rate (in decimal form),
- $ t $ is the time in years,
- $ e $ is the base of the natural logarithm (approximately 2.71828).

### Step 1: Write down the known quantities
- Interest rate $ r = 16\% = 0.16 $,
- After 13 years, the balance is $70,000 (\text{A} = 70,000)$,
- After 5 years, an amount equal to 20% of the original investment is withdrawn.

We need to find the amount of the withdrawal. First, let's solve for the original investment $ P $.

### Step 2: Calculate the original investment
The account was untouched for the first 5 years, so the amount at the end of 5 years (before the withdrawal) is:

\[
A_5 = P e^{0.16 \times 5} = P e^{0.8}
\]

At the end of 13 years, the balance was $70,000, after a withdrawal of 20% of the original investment at year 5. This means that the principal remaining after the withdrawal, which is $ P - 0.2P = 0.8P $, grows for 8 more years (from year 5 to year 13):

\[
70,000 = 0.8P \cdot e^{0.16 \times 8} = 0.8P \cdot e^{1.28}
\]

### Step 3: Solve for $ P $
Now, solve for $ P $:

\[
70,000 = 0.8P \cdot e^{1.28}
\]

\[
P = \frac{70,000}{0.8 \cdot e^{1.28}}
\]

Let me calculate this value.The original investment $ P $ was approximately \$24,328.26.

### Step 4: Calculate the withdrawal
The withdrawal was 20% of the original investment, so:

\[
\text{Withdrawal} = 0.2 \times P = 0.2 \times 24,328.26
\]

Let me calculate the amount of the withdrawal.The amount of the withdrawal is approximately \$4,865.65.

Would you like more details or have any questions?

Here are 5 related questions:
1. How is continuously compounded interest different from simple interest?
2. What would the balance be if no withdrawal had been made after 5 years?
3. How would the answer change if the interest rate were different?
4. What is the formula for compounding at other intervals, such as quarterly or monthly?
5. How can the time to double an investment be calculated using continuous compounding?

**Tip:** Always remember that continuous compounding grows investments at a faster rate compared to other compounding methods due to the constant application of interest.

Solve a problem involving continuous compounding at 16% per annum. Calculate the original investment and withdrawal amount after 5 years, given that the balance after 13 years is $70,000. This problem covers exponential growth and interest calculation concepts suitable for high school students.

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