To determine which option has the greater rate of change between years 9 and 10, we need to evaluate both functions \( p(x) \) and \( q(x) \) at \( x = 9 \) and \( x = 10 \).
Option #1: \[ p(x) = 5000(1.06)^x \]
Calculating for \( x = 9 \): \[ p(9) = 5000(1.06)^9 \] Using a calculator for \( (1.06)^9 \): \[ (1.06)^9 \approx 1.59385 \] So, \[ p(9) \approx 5000 \times 1.59385 \approx 7969.25 \]
Calculating for \( x = 10 \): \[ p(10) = 5000(1.06)^{10} \] Using a calculator for \( (1.06)^{10} \): \[ (1.06)^{10} \approx 1.68948 \] So, \[ p(10) \approx 5000 \times 1.68948 \approx 8447.41 \]
Now calculating the rate of change between years 9 and 10 for Option #1: \[ \text{Rate of change} = p(10) - p(9) \approx 8447.41 - 7969.25 \approx 478.16 \]
Option #2: \[ q(x) = 500x + 5000 \]
Calculating for \( x = 9 \): \[ q(9) = 500(9) + 5000 = 4500 + 5000 = 9500 \]
Calculating for \( x = 10 \): \[ q(10) = 500(10) + 5000 = 5000 + 5000 = 10000 \]
Now calculating the rate of change between years 9 and 10 for Option #2: \[ \text{Rate of change} = q(10) - q(9) = 10000 - 9500 = 500 \]
Comparison of Rates of Change:
- Rate of change for Option #1 (p): \( 478.16 \)
- Rate of change for Option #2 (q): \( 500 \)
Since \( 500 > 478.16 \), Option #2 has the greater rate of change between years 9 and 10.
Final Decision: If Miguel plans to invest his money for only 10 years, he should choose Option #2.
Thus, we conclude:
Option 2 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 2.
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