Question 6 (Essay Worth 4 points) (An Interest in Growth Models HC) A principal amount of 5,000 is placed in a savings account with 5% annual interest compounded quarterly. Part A: List the total account balances for years 0 through 2 Show all necessary work.(2 points) Part B: Which type of function best models the data? (1 point) Part C: Solve for the APY. Show all necessary work. (1 point)

Part A: To find the total account balances for years 0 through 2, we can use the formula for compound interest:<br /><br />\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]<br /><br />where:<br />- \( A \) is the amount of money accumulated after \( n \) years, including interest.<br />- \( P \) is the principal amount (the initial amount of money).<br />- \( r \) is the annual interest rate (in decimal).<br />- \( n \) is the number of times that interest is compounded per year.<br />- \( t \) is the time the money is invested for in years.<br /><br />Given:<br />- \( P = \$5,000 \)<br />- \( r = 5\% = 0.05 \)<br />- \( n = 4 \) (since the interest is compounded quarterly)<br />- \( t = 0, 1, 2 \) (for years 0 through 2)<br /><br />Let's calculate the total account balances for each year:<br /><br />For year 0:<br />\[ A = 5000 \left(1 + \frac{0.05}{4}\right)^{4 \cdot 0} = 5000 \left(1 + 0.0125\right)^0 = 5000 \cdot 1 = 5000 \]<br /><br />For year 1:<br />\[ A = 5000 \left(1 + \frac{0.05}{4}\right)^{4 \cdot 1} = 5000 \left(1 + 0.0125\right)^4 \approx 5000 \cdot 1.050945 = 5254.75 \]<br /><br />For year 2:<br />\[ A = 5000 \left(1 + \frac{0.05}{4}\right)^{4 \cdot 2} = 5000 \left(1 + 0.0125\right)^8 \approx 5000 \cdot 1.104622 = 5523.11 \]<br /><br />So, the total account balances for years 0 through 2 are:<br />- Year 0: \$5,000<br />- Year 1: \$5,254.75<br />- Year 2: \$5,523.11<br /><br />Part B: The type of function that best models the data is an exponential function. This is because the amount of money in the account grows exponentially over time due to the effect of compounding interest.<br /><br />Part C: To solve for the APY (Annual Percentage Yield), we can use the formula:<br /><br />\[ APY = \left(1 + \frac{r}{n}\right)^n - 1 \]<br /><br />Given:<br />- \( r = 0.05 \)<br />- \( n = 4 \)<br /><br />Let's calculate the APY:<br /><br />\[ APY = \left(1 + \frac{0.05}{4}\right)^4 - 1 \approx 0.050945 \]<br /><br />So, the APY is approximately 5.0945%.