question 15 (essay worth 10 points) (11c) two people are placing a principal amount of $3,425 in a savings…

Explanation:

Part A:

Step1: Identify function for Account A

Account A has quarterly - compounded interest. The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $P$ is the principal, $r$ is the annual interest rate, $n$ is the number of times compounded per year, and $t$ is the number of years. This is an exponential function of the form $y = a(b)^x$ where $a = P$, $b=(1 +\frac{r}{n})^n$ and $x = t$. So, an exponential function models Account A.

Step2: Identify function for Account B

Account B has continuously - compounded interest. The formula is $A=Pe^{rt}$, which is also an exponential function (since the base of the natural exponential function $e\approx2.718$). So, an exponential function models Account B.

Part B:

Step1: Recall APY formula for compound - interest

The APY (Annual Percentage Yield) formula for compound interest is $APY=(1 +\frac{r}{n})^n-1$, where $r$ is the annual interest rate and $n$ is the number of compounding periods per year. For Account A, $r = 0.0471$ and $n = 4$ (quarterly compounding).

Step2: Substitute values into the formula

$APY=(1+\frac{0.0471}{4})^4 - 1$. First, calculate $\frac{0.0471}{4}=0.011775$. Then $1+\frac{0.0471}{4}=1.011775$. Next, $(1.011775)^4\approx1.0479$. So, $APY = 1.0479-1=0.0479$ or $4.79%$.

Part C:

Step1: Recall APY formula for continuous - compounding

The formula for APY in continuous compounding is $APY = e^r-1$, where $r$ is the annual interest rate. Given $r = 0.0471$.

Step2: Calculate APY

$APY=e^{0.0471}-1$. Since $e^{0.0471}\approx1.0482$, then $APY=1.0482 - 1=0.0482$ or $4.82%$.

Answer:

Part A:

Both Account A and Account B are best modeled by exponential functions. Account A uses the compound - interest formula $A = P(1+\frac{r}{n})^{nt}$ and Account B uses the continuous - compounding formula $A = Pe^{rt}$, both of which are forms of exponential functions.

Part B:

The APY for Account A is approximately $4.79%$.

Part C:

The APY for Account B is approximately $4.82%$.