Question

The graph shows the total amount of an investment account that has a principal of $5,000 and increases by 70% every 10 years. Estimate the balance of the investment after 15 years to the nearest thousand.

Answer 1

To estimate the balance of the investment after 15 years, we first need to understand how the investment grows.

Given that the principal is $5,000 and it increases by 70% every 10 years, we can calculate the amount after 10 years:

1. **Calculate the amount after 10 years:**
- The increase is 70% of $5,000, which can be calculated as follows:
\[
70\% \text{ of } 5000 = 0.70 \times 5000 = 3500
\]
- Therefore, the total amount after 10 years is:
\[
5000 + 3500 = 8500
\]

2. **Calculate the amount after another 5 years:**
- To estimate the balance after 15 years, we can find the annual growth rate based on the principal amount of $5,000 increasing to $8,500 over 10 years.
- The growth from $5,000 to $8,500 constitutes a growth factor of:
\[
\text{Growth factor after 10 years} = \frac{8500}{5000} = 1.7
\]

- The annual growth rate (r) can be determined using the formula for compound growth:
\[
(1 + r)^{10} = 1.7
\]
- Taking the 10th root:
\[
1 + r \approx 1.7^{(1/10)}
\]
- Using a calculator:
\[
r \approx 1.0547 \text{ (approximately)}
\]

This indicates an annual growth rate of about 5.47%.

3. **Calculate the total amount after 15 years:**
- Since we have already computed the amount after 10 years as $8,500, we can compute the amount for the additional 5 years:
\[
\text{Amount after 15 years} = 8500 \times (1 + r)^5
\]
\[
= 8500 \times (1.0547)^5
\]
- Calculating \((1.0547)^5\) using a calculator:
\[
(1.0547)^5 \approx 1.302
\]
- Therefore:
\[
\text{Amount after 15 years} \approx 8500 \times 1.302 \approx 11067
\]

Finally, rounding to the nearest thousand, the estimated balance of the investment after 15 years is approximately **$11,000**.