To determine which option has a greater rate of change between years 9 and 10, let’s analyze both functions.
Option #1: Function p(x) The function for option #1 is given by: \[ p(x) = 5,000(1.06)^x \]
To find the values at years 9 and 10:
- \( p(9) = 5,000(1.06)^9 \)
- \( p(10) = 5,000(1.06)^{10} \)
Calculating these values:
- \( p(9) = 5,000(1.06)^9 \approx 5,000(1.6895) \approx 8,447.50 \)
- \( p(10) = 5,000(1.06)^{10} \approx 5,000(1.7908) \approx 8,954.00 \)
Now calculate the rate of change between years 9 and 10 for option #1: \[ \text{Rate of Change} = p(10) - p(9) = 8,954.00 - 8,447.50 = 506.50 \]
Option #2: Function q(x) The function for option #2 is given by: \[ q(x) = 500x + 5,000 \]
To find the values at years 9 and 10:
- \( q(9) = 500(9) + 5,000 = 4,500 + 5,000 = 9,500 \)
- \( q(10) = 500(10) + 5,000 = 5,000 + 5,000 = 10,000 \)
Now calculate the rate of change between years 9 and 10 for option #2: \[ \text{Rate of Change} = q(10) - q(9) = 10,000 - 9,500 = 500 \]
Comparison of Rate of Change
- Rate of change for Option #1: 506.50
- Rate of change for Option #2: 500
Between years 9 and 10, Option #1 has the greater rate of change (506.50 > 500).
Conclusion:
- The correct statement is: Option #1 has the greater rate of change between years 9 and 10.
- Since Miguel plans to withdraw after 10 years, he should choose Option #1 as it provides more growth in that time frame.