Question

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. 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?
Option # ____ 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 # ____

Answer 1

To compare the rates of change between the two investment options, we need to calculate the functions at years 9 and 10 and then determine the rate of change for each option.

### Option #1: Exponential Growth
The function for Option #1 is:
\[ p(x) = 5,000(1.06)^x \]

Calculating for years 9 and 10:

1. For year 9:
\[
p(9) = 5,000(1.06)^9
\]

Approximating \( (1.06)^9 \):
\[
(1.06)^9 \approx 1.715
\]
Thus,
\[
p(9) \approx 5,000 \times 1.715 \approx 8,575
\]

2. For year 10:
\[
p(10) = 5,000(1.06)^{10}
\]

Approximating \( (1.06)^{10} \):
\[
(1.06)^{10} \approx 1.822
\]
Thus,
\[
p(10) \approx 5,000 \times 1.822 \approx 9,110
\]

Now, calculate the rate of change between years 9 and 10:
\[
\text{Rate of Change for Option #1} = p(10) - p(9) = 9,110 - 8,575 = 535
\]

### Option #2: Linear Growth
The function for Option #2 is:
\[ q(x) = 500x + 5,000 \]

Calculating for years 9 and 10:

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

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

Now, calculate the rate of change between years 9 and 10:
\[
\text{Rate of Change for Option #2} = q(10) - q(9) = 10,000 - 9,500 = 500
\]

### Comparison of Rates of Change
- Rate of Change for Option #1: 535
- Rate of Change for Option #2: 500

### Conclusion
Since Option #1 has a greater rate of change between years 9 and 10, the answer is:

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.