Question

Use the Table to answer the question
team number 1 2 3 4 5 6 7 8 9 10
sequence for option #1 5,300.5618.
sequence for option #2
(1 point)
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 determine which option has a greater rate of change between years 9 and 10, we need to calculate the values of both functions \( p(x) \) and \( q(x) \) for \( x = 9 \) and \( x = 10 \).

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

Calculating \( p(9) \) and \( p(10) \):
- \( p(9) = 5,000(1.06)^9 \)
- \( p(10) = 5,000(1.06)^{10} \)

Calculating the values:
- \( 1.06^9 \approx 1.6895 \) (Using a calculator)
- \( 1.06^{10} \approx 1.7908 \)

Now, substituting these back into the equation:
- \( p(9) \approx 5,000 \times 1.6895 \approx 8,447.50 \)
- \( p(10) \approx 5,000 \times 1.7908 \approx 8,954.00 \)

Now we calculate the rate of change between years 9 and 10:
\[
\text{Rate of change} = p(10) - p(9) \approx 8,954.00 - 8,447.50 \approx 506.50
\]

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

Calculating \( q(9) \) and \( q(10) \):
- \( q(9) = 500 \times 9 + 5,000 = 4,500 + 5,000 = 9,500 \)
- \( q(10) = 500 \times 10 + 5,000 = 5,000 + 5,000 = 10,000 \)

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

### Summary of Rates of Change:
- Rate of change for Option #1 between years 9 and 10: **506.50**
- Rate of change for Option #2 between years 9 and 10: **500**

### Conclusion:
Option #1 has the greater rate of change between years 9 and 10. Therefore, if Miguel plans to invest his money for only 10 years, he should choose **Option #1**.

Final answers:
- 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.