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 determine which option has the greater rate of change between years 9 and 10, we need to evaluate both functions \( p(x) \) and \( q(x) \) at \( x = 9 \) and \( x = 10 \).

**Option #1:**
\[ p(x) = 5000(1.06)^x \]

Calculating for \( x = 9 \):
\[ p(9) = 5000(1.06)^9 \]
Using a calculator for \( (1.06)^9 \):
\[ (1.06)^9 \approx 1.59385 \]
So,
\[ p(9) \approx 5000 \times 1.59385 \approx 7969.25 \]

Calculating for \( x = 10 \):
\[ p(10) = 5000(1.06)^{10} \]
Using a calculator for \( (1.06)^{10} \):
\[ (1.06)^{10} \approx 1.68948 \]
So,
\[ p(10) \approx 5000 \times 1.68948 \approx 8447.41 \]

Now calculating the rate of change between years 9 and 10 for Option #1:
\[ \text{Rate of change} = p(10) - p(9) \approx 8447.41 - 7969.25 \approx 478.16 \]

**Option #2:**
\[ q(x) = 500x + 5000 \]

Calculating for \( x = 9 \):
\[ q(9) = 500(9) + 5000 = 4500 + 5000 = 9500 \]

Calculating for \( x = 10 \):
\[ q(10) = 500(10) + 5000 = 5000 + 5000 = 10000 \]

Now calculating the rate of change between years 9 and 10 for Option #2:
\[ \text{Rate of change} = q(10) - q(9) = 10000 - 9500 = 500 \]

**Comparison of Rates of Change:**
- Rate of change for Option #1 (p): \( 478.16 \)
- Rate of change for Option #2 (q): \( 500 \)

Since \( 500 > 478.16 \), Option #2 has the greater rate of change between years 9 and 10.

**Final Decision:**
If Miguel plans to invest his money for only 10 years, he should choose Option #2.

Thus, we conclude:

Option **2** 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 **2**.