The amount of money, A, earned after being compounded monthly for t years with Hamilton Credit Union is represented by the function f(t)=10,000(1.009)12t

Question

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Which of the following statements best describes the initial amount invested and its constant percent rate of change?

(1 point)
Responses

The initial amount of money invested is $1,009, which grows at an annual rate of about 11%.
The initial amount of money invested is $1,009, which grows at an annual rate of about 11%.

The initial amount of money invested is $10,000, which grows at an annual rate of about 11%.
The initial amount of money invested is $10,000, which grows at an annual rate of about 11%.

The initial amount of money invested is $10,000, which decays at an annual rate of about 9%.
The initial amount of money invested is $10,000, which decays at an annual rate of about 9%.

The initial amount of money invested is $10,000, which grows at an annual rate of about 9%.
The initial amount of money invested is $10,000, which grows at an annual rate of about 9%.
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Answer 1

To analyze the function provided and determine the correct statement, let's break down the function \( f(t) = 10,000(1.009)^{12t} \).

1. Initial Investment: The initial amount invested is represented by the coefficient of the function at \( t = 0 \), which is \( 10,000 \).

2. Growth Rate: The term \( (1.009)^{12t} \) indicates monthly compounding. The monthly growth factor is \( 1.009 \). To find the annual growth rate, we can convert this monthly growth factor to the equivalent annual growth rate:

   \[ \text{Annual Growth Rate} = (1.009)^{12} - 1 \] Now calculate \( (1.009)^{12} \):

   \[ (1.009)^{12} \approx 1.1136 \]

   Thus, the annual growth rate is approximately:

   \[ 1.1136 - 1 \approx 0.1136 \text{ or } 11.36% \]

Given this analysis, we can conclude that:

• The initial amount of money invested is $10,000.
• The account grows at an annual rate of about 11%.

Therefore, the best response that describes the initial amount invested and its constant percent rate of change is:

The initial amount of money invested is $10,000, which grows at an annual rate of about 11%.