To find the option with the greater rate of change between years 9 and 10, we need to evaluate the two functions \( p(x) \) and \( q(x) \) at \( x = 9 \) and \( x = 10 \).
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Option #1: \( p(x) = 5,000(1.06)^x \)
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For \( x = 9 \): \[ p(9) = 5,000(1.06)^9 \] Calculating \( (1.06)^9 \): \[ (1.06)^9 \approx 1.715 \] Therefore, \[ p(9) \approx 5,000 \times 1.715 \approx 8,575 \]
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For \( x = 10 \): \[ p(10) = 5,000(1.06)^{10} \] Calculating \( (1.06)^{10} \): \[ (1.06)^{10} \approx 1.821 \] Therefore, \[ p(10) \approx 5,000 \times 1.821 \approx 9,105 \]
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The rate of change between years 9 and 10 for Option #1 is: \[ p(10) - p(9) \approx 9,105 - 8,575 \approx 530 \]
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Option #2: \( q(x) = 500x + 5,000 \)
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For \( x = 9 \): \[ q(9) = 500(9) + 5,000 = 4,500 + 5,000 = 9,500 \]
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For \( x = 10 \): \[ q(10) = 500(10) + 5,000 = 5,000 + 5,000 = 10,000 \]
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The rate of change between years 9 and 10 for Option #2 is: \[ q(10) - q(9) = 10,000 - 9,500 = 500 \]
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Now we compare the rates of change:
- Rate of change for Option #1 between years 9 and 10: 530
- Rate of change for Option #2 between years 9 and 10: 500
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 as it offers a higher rate of growth.
Final answer:
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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