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.
Answers
GPT-4o mini
To estimate the balance of an investment that grows by 70% every 10 years, we can start with the initial principal of $5,000.
1. **Finding the growth factor**: An increase of 70% means the amount becomes \( 1 + 0.70 = 1.70 \) times the original amount every 10 years.
2. **Calculating the amount after 10 years**:
\[
\text{Amount after 10 years} = 5000 \times 1.70 = 8500
\]
3. **Calculating the amount after 15 years**: After 15 years, which is one and a half periods of 10 years, we can apply the growth factor accordingly.
- The 5-year period will involve taking the amount after 10 years and applying a growth of 70% over a half period. Since 70% growth over a full period (10 years) equates to multiplying by 1.70, we can find the 5-year growth with a growth factor of \( (1.70)^{0.5} \):
\[
(1.70)^{0.5} \approx 1.3038 \quad (\text{taking the square root of the growth for half the period})
\]
- Now, apply this factor to the amount after 10 years:
\[
\text{Amount after 15 years} = 8500 \times (1.70)^{0.5} \approx 8500 \times 1.3038 \approx 11083.3
\]
4. **Rounding off**: To the nearest thousand, we round 11083.3 to 11000.
So, the estimated balance of the investment after 15 years is **$11,000**.
1. **Finding the growth factor**: An increase of 70% means the amount becomes \( 1 + 0.70 = 1.70 \) times the original amount every 10 years.
2. **Calculating the amount after 10 years**:
\[
\text{Amount after 10 years} = 5000 \times 1.70 = 8500
\]
3. **Calculating the amount after 15 years**: After 15 years, which is one and a half periods of 10 years, we can apply the growth factor accordingly.
- The 5-year period will involve taking the amount after 10 years and applying a growth of 70% over a half period. Since 70% growth over a full period (10 years) equates to multiplying by 1.70, we can find the 5-year growth with a growth factor of \( (1.70)^{0.5} \):
\[
(1.70)^{0.5} \approx 1.3038 \quad (\text{taking the square root of the growth for half the period})
\]
- Now, apply this factor to the amount after 10 years:
\[
\text{Amount after 15 years} = 8500 \times (1.70)^{0.5} \approx 8500 \times 1.3038 \approx 11083.3
\]
4. **Rounding off**: To the nearest thousand, we round 11083.3 to 11000.
So, the estimated balance of the investment after 15 years is **$11,000**.