To determine which option has a greater rate of change between years 9 and 10, we need to calculate the values of both functions \( p(x) \) and \( q(x) \) for \( x = 9 \) and \( x = 10 \).
Option #1:
The function for Option #1 is given by: \[ p(x) = 5,000(1.06)^x \]
Calculating \( p(9) \) and \( p(10) \):
- \( p(9) = 5,000(1.06)^9 \)
- \( p(10) = 5,000(1.06)^{10} \)
Calculating the values:
- \( 1.06^9 \approx 1.6895 \) (Using a calculator)
- \( 1.06^{10} \approx 1.7908 \)
Now, substituting these back into the equation:
- \( p(9) \approx 5,000 \times 1.6895 \approx 8,447.50 \)
- \( p(10) \approx 5,000 \times 1.7908 \approx 8,954.00 \)
Now we calculate the rate of change between years 9 and 10: \[ \text{Rate of change} = p(10) - p(9) \approx 8,954.00 - 8,447.50 \approx 506.50 \]
Option #2:
The function for Option #2 is given by: \[ q(x) = 500x + 5,000 \]
Calculating \( q(9) \) and \( q(10) \):
- \( q(9) = 500 \times 9 + 5,000 = 4,500 + 5,000 = 9,500 \)
- \( q(10) = 500 \times 10 + 5,000 = 5,000 + 5,000 = 10,000 \)
Now we calculate the rate of change between years 9 and 10: \[ \text{Rate of change} = q(10) - q(9) = 10,000 - 9,500 = 500 \]
Summary of Rates of Change:
- Rate of change for Option #1 between years 9 and 10: 506.50
- Rate of change for Option #2 between years 9 and 10: 500
Conclusion:
Option #1 has the greater rate of change between years 9 and 10. Therefore, if Miguel plans to invest his money for only 10 years, he should choose Option #1.
Final answers:
- Option #1 has the greater rate of change between years 9 and 10.
- If Miguel plans to invest his money for only 10 years, he should choose Option #1.
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