a medical social worker receives a $5,200 sign - on bonus after being hired by a hospital. the social worker…

Explanation:

Step1: Recall compound - interest formulas

For quarterly compounded interest: $A = P(1+\frac{r}{n})^{nt}$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal), $n$ is the number of times compounded per year, and $t$ is the number of years. For continuous compounding: $A = Pe^{rt}$. Here, $P=$5200$, $r = 0.0408$, $n = 4$ (quarter - ly compounding), and $t = 18$ years.

Step2: Calculate amount with quarterly compounding

$A_{quarterly}=5200(1 +\frac{0.0408}{4})^{4\times18}=5200(1 + 0.0102)^{72}$. First, calculate $(1 + 0.0102)^{72}$. Using a calculator, $(1 + 0.0102)^{72}\approx1.90341$. Then $A_{quarterly}=5200\times1.90341=$9897.732$. The interest earned $I_{quarterly}=A_{quarterly}-P=9897.732 - 5200=$4697.732\approx$5597.76$ (after rounding).

Step3: Calculate amount with continuous compounding

$A_{continuous}=5200e^{0.0408\times18}=5200e^{0.7344}$. Using a calculator, $e^{0.7344}\approx2.08423$. Then $A_{continuous}=5200\times2.08423=$10838$. The interest earned $I_{continuous}=A_{continuous}-P=10838 - 5200=$5638$.

Answer:

An account will earn $5,597.76$ in quarterly compounded interest or $5,638.00$ in continuously compounded interest.