Question

Use the table to answer the question.



(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 find the option with the greater rate of change between years 9 and 10, we need to evaluate the two functions \( p(x) \) and \( q(x) \) at \( x = 9 \) and \( x = 10 \).

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

- For \( x = 9 \):
\[
p(9) = 5,000(1.06)^9
\]
Calculating \( (1.06)^9 \):
\[
(1.06)^9 \approx 1.715
\]
Therefore,
\[
p(9) \approx 5,000 \times 1.715 \approx 8,575
\]

- For \( x = 10 \):
\[
p(10) = 5,000(1.06)^{10}
\]
Calculating \( (1.06)^{10} \):
\[
(1.06)^{10} \approx 1.821
\]
Therefore,
\[
p(10) \approx 5,000 \times 1.821 \approx 9,105
\]

- The rate of change between years 9 and 10 for Option #1 is:
\[
p(10) - p(9) \approx 9,105 - 8,575 \approx 530
\]

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

- For \( x = 9 \):
\[
q(9) = 500(9) + 5,000 = 4,500 + 5,000 = 9,500
\]

- For \( x = 10 \):
\[
q(10) = 500(10) + 5,000 = 5,000 + 5,000 = 10,000
\]

- The rate of change between years 9 and 10 for Option #2 is:
\[
q(10) - q(9) = 10,000 - 9,500 = 500
\]

Now we compare the rates of change:

- Rate of change for Option #1 between years 9 and 10: **530**
- Rate of change for Option #2 between years 9 and 10: **500**

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** as it offers a higher rate of growth.

Final answer:

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**.