Miguel is planning to invest $5,000. He plans to withdraw his investment and earnings in approximately 10 years. With Option #1, his money will grow according to the function p(x)=5,000(1.06)^x

Question

. With Option #2 his money will grow according to the function q(x)=500x+5,000
. He creates sequences for both functions. Which option has the greater rate of change between years 9 and 10? Which option should he choose for his money?

Answer 1

To compare the rate of change between years 9 and 10 for both investment options, we first need to evaluate the functions \( p(x) \) and \( q(x) \) at those specific years.

Option #1: \( p(x) = 5,000(1.06)^x \)

Calculate \( p(9) \) and \( p(10) \):

1. Calculate \( p(9) \): \[ p(9) = 5,000(1.06)^9 \] Using the calculator: \[ (1.06)^9 \approx 1.689478 \] So, \[ p(9) \approx 5,000 \times 1.689478 \approx 8,447.39 \]

2. Calculate \( p(10) \): \[ p(10) = 5,000(1.06)^{10} \] Using the calculator: \[ (1.06)^{10} \approx 1.790847 \] So, \[ p(10) \approx 5,000 \times 1.790847 \approx 8,954.24 \]

3. Calculate the rate of change from year 9 to year 10: \[ \text{Rate of change for } p = p(10) - p(9) \approx 8,954.24 - 8,447.39 \approx 506.85 \]

Option #2: \( q(x) = 500x + 5,000 \)

Calculate \( q(9) \) and \( q(10) \):

1. Calculate \( q(9) \): \[ q(9) = 500 \times 9 + 5,000 = 4,500 + 5,000 = 9,500 \]

2. Calculate \( q(10) \): \[ q(10) = 500 \times 10 + 5,000 = 5,000 + 5,000 = 10,000 \]

3. Calculate the rate of change from year 9 to year 10: \[ \text{Rate of change for } q = q(10) - q(9) = 10,000 - 9,500 = 500 \]

Comparing the Rates of Change

• Rate of change for Option #1 (\( p \)) = 506.85
• Rate of change for Option #2 (\( q \)) = 500

Conclusion

Between years 9 and 10, Option #1 has a greater rate of change (506.85) compared to Option #2 (500).

Recommendation:

Miguel should choose Option #1, as it offers a higher rate of growth for his investment.