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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)

1 answer

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).