Bhongo received R32 412,87 after investing an amount of money in an account earning interest at a continuous compounding rate of 10,15% per year. The amount of money that he invested 57 weeks earlier, is approximately

a.

R29 000,00.

b.

R29 153,86.

c.

R32 768,16.

d.

R36 227,38.

To determine the amount Bhongo originally invested, given that he received R32,412.87 after 57 weeks with a continuous compounding interest rate of 10.15% per year, we can use the formula for continuous compounding:

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

Where:
- \( A \) is the amount of money accumulated after the interest is applied (R32,412.87).
- \( P \) is the principal amount (the initial amount of money invested).
- \( r \) is the annual continuous interest rate (10.15% or 0.1015).
- \( t \) is the time the money is invested, in years.

First, convert 57 weeks to years:
\[
t = \frac{57}{52} \approx 1.0962 \text{ years}
\]

Now, we can plug in the values and solve for \( P \):

\[
32,412.87 = P \cdot e^{0.1015 \times 1.0962}
\]

First, calculate the exponent:

\[
0.1015 \times 1.0962 \approx 0.1113
\]

Now, calculate \( e^{0.1113} \):

\[
e^{0.1113} \approx 1.1177
\]

Now, solve for \( P \):

\[
P = \frac{32,412.87}{1.1177} \approx 29,000.14
\]

The amount of money Bhongo originally invested is approximately **R29,000.14**.

So, the closest answer is:

**a. R29 000,00.**

Would you like more details or have any questions?

Here are some related questions you might be interested in:
1. How would the answer change if the interest rate was compounded annually instead of continuously?
2. What is the formula for continuous compounding, and how does it differ from regular compounding?
3. How would the original investment amount change if the interest rate were 12% instead of 10.15%?
4. If Bhongo wanted to reach R40,000 instead of R32,412.87 in 57 weeks, how much should he have invested?
5. What are some common scenarios where continuous compounding is used in finance?

**Tip:** Continuous compounding assumes interest is added an infinite number of times per year, making it a useful model for maximizing returns over time.

Learn how to calculate the initial investment amount using continuous compounding with a detailed step-by-step solution. This problem involves continuous compounding at a rate of 10.15% per year over 57 weeks. Suitable for university-level students studying finance or mathematics.

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