Questions: 7,400 is invested in an account earning 6.8% interest (APR), compounded daily. Write a function showing the value of the account after t years, where the annual growth rate can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage of growth per year (APY), to the nearest hundredth of a percent.

7,400 is invested in an account earning 6.8% interest (APR), compounded daily. Write a function showing the value of the account after t years, where the annual growth rate can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage of growth per year (APY), to the nearest hundredth of a percent.
Transcript text: $7,400$ is invested in an account earning $6.8 \%$ interest (APR), compounded daily. Write a function showing the value of the account after $t$ years, where the annual growth rate can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage of growth per year (APY), to the nearest hundredth of a percent.
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Solution

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Solution Steps

Step 1: Calculate Future Value

To find the future value \( A \) of the investment after \( t = 5 \) years, we use the formula for compound interest:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

Substituting the values \( P = 7400 \), \( r = 0.068 \), \( n = 365 \), and \( t = 5 \):

\[ A = 7400 \left(1 + \frac{0.068}{365}\right)^{365 \times 5} \approx 10396.2829 \]

Thus, the future value of the account after 5 years is approximately \( 10396.2829 \).

Step 2: Calculate Annual Percentage Yield (APY)

The annual percentage yield (APY) is calculated using the formula:

\[ \text{APY} = \left(1 + \frac{r}{n}\right)^n - 1 \]

Substituting the values \( r = 0.068 \) and \( n = 365 \):

\[ \text{APY} = \left(1 + \frac{0.068}{365}\right)^{365} - 1 \approx 0.0704 \]

Converting this to a percentage gives us \( \text{APY} \approx 7.04\% \).

Final Answer

The future value of the account after 5 years is approximately \( 10396.2829 \), and the annual percentage yield (APY) is approximately \( 7.04\% \).

\[ \boxed{A \approx 10396.2829, \text{ APY } \approx 7.04\%} \]

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