find the interest earned on a $30,000 deposit for six months at $1\frac{1}{2}%$ interest, compounded…

Explanation:

Step1: Identify the formula for continuous - compounding

The formula for continuous - compounding is $A = Pe^{rt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (in decimal form), and $t$ is the time in years. The interest earned $I=A - P$. So, $I = Pe^{rt}-P=P(e^{rt}-1)$.

Step2: Convert the given values to the correct form

The principal $P = 30000$, the annual interest rate $r=1.5%=0.015$, and the time $t=\frac{6}{12}=0.5$ years.

Step3: Substitute the values into the formula

Substitute $P = 30000$, $r = 0.015$, and $t = 0.5$ into the formula $I=P(e^{rt}-1)$. $I = 30000(e^{0.015\times0.5}-1)$. First, calculate the exponent: $0.015\times0.5 = 0.0075$. Then, find $e^{0.0075}$. Using a calculator, $e^{0.0075}\approx1.007528$. Next, $e^{0.0075}-1\approx1.007528 - 1=0.007528$. Finally, $I = 30000\times0.007528 = 225.84$.

Answer:

$225.84$