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
(2 points)
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)*. With Option #2 his money will grow according to the function
9(x) = 500х + 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 # i 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

Rate of change between years 9 and 10:

- Option 1: p(x) = 5000(1.06)^x
p(10) − p(9) = 5000[(1.06)^10 − (1.06)^9] = 5000(1.06^9)(0.06) ≈ 5000(1.68948)(0.06) ≈ $506.84

- Option 2: g(x) = 500x + 5000
g(10) − g(9) = 500(10 − 9) = $500

So Option 1 has the greater rate of change between years 9 and 10 (≈ $506.84 vs $500).

However, after 10 years:
- p(10) ≈ 5000(1.06^10) ≈ $8,954.23
- g(10) = 500(10) + 5000 = $10,000

Since Miguel plans to withdraw in about 10 years, he should choose Option 2 because it gives the larger final amount.