Questions: Write a function to represent the value A of the necklace after t years, assuming the initial value of 1. Then write an equivalent function with a power of 12t.

Solution Steps

To solve this problem, we need to create two functions to represent the value of the necklace over time. The first function will use the annual growth rate, and the second function will convert this to a monthly growth rate and use a power of \(12t\) to represent the value over time.

1. Annual Growth Rate Function: The value of the necklace increases by 3.2% per year. We can represent this with the formula \(A(t) = 1 \times (1 + 0.032)^t\), where \(t\) is the number of years.
2. Monthly Growth Rate Function: To convert the annual growth rate to a monthly growth rate, we need to find the equivalent monthly rate that, when compounded over 12 months, gives the same annual growth. This can be done using the formula \((1 + r_{annual}) = (1 + r_{monthly})^{12}\). Solving for \(r_{monthly}\), we get \(r_{monthly} = (1 + 0.032)^{1/12} - 1\). The function then becomes \(A(t) = 1 \times (1 + r_{monthly})^{12t}\).

The value of the necklace after \( t \) years, assuming an initial value of \( \$1 \) and an annual growth rate of \( 3.2\% \), can be expressed as: \[ A(t) = 1 \times (1 + 0.032)^t \] For \( t = 5 \): \[ A(5) = 1 \times (1 + 0.032)^5 \approx 1.1706 \]

To find the equivalent monthly growth rate, we first calculate: \[ r_{monthly} = (1 + 0.032)^{\frac{1}{12}} - 1 \] Using this monthly growth rate, the value of the necklace after \( t \) years can be expressed as: \[ A(t) = 1 \times (1 + r_{monthly})^{12t} \] For \( t = 5 \): \[ A(5) \approx 1.1706 \]

Final Answer