Question

Use the table to answer the question.

Term Number 1 2 3 4 5 6 7 8 9 10
Sequence for
Option #1 5,300 5,618 5,955.1 6,312.4 6,691.1 7,092.6 7,518.2 7,969.2 8,447.4 8,954.2
Sequence for
Option #2 5,500 6,000 6,500 7,000 7,500 8,000 8,500 9,000 9,500 10,000
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⋅1.06x
. 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?

(1 point)

Answer 1

To determine which investment option has a greater rate of change between years 9 and 10, we need to calculate the rate of change for both sequences at those specific terms.

### For Option #1:
From the table, the values for Option #1 are:
- Year 9: \( p(9) = 8,447.4 \)
- Year 10: \( p(10) = 8,954.2 \)

#### Rate of change for Option #1:
\[
\text{Rate of change} = p(10) - p(9) = 8,954.2 - 8,447.4 = 506.8
\]

### For Option #2:
From the table, the values for Option #2 are:
- Year 9: \( q(9) = 9,500 \)
- Year 10: \( q(10) = 10,000 \)

#### Rate of change for Option #2:
\[
\text{Rate of change} = q(10) - q(9) = 10,000 - 9,500 = 500
\]

### Comparing Rates of Change:
- Rate of change for Option #1 (between years 9 and 10): \( 506.8 \)
- Rate of change for Option #2 (between years 9 and 10): \( 500 \)

### Conclusion:
Since the rate of change for Option #1 is greater than that for Option #2 (\( 506.8 > 500 \)), Miguel should choose **Option #1**, as it provides a higher rate of growth in the final year (between years 9 and 10).