Question

note: please make sure to properly format your answers. all dollar figures in the answers need to include the d over 1,000 should include the comma ($2,354.67). all percentage values in the answers need to include a perce without specific rounding instructions, round your answers to two decimal places, show both decimal places (5. find the interest earned on a $30,000 deposit for six months at 1\\(\frac{1}{2}\\)% interest, compounded continuously.

Explanation:

Step1: Recall the formula for continuous compounding

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

First, convert the interest rate and time to the appropriate units.

The interest rate is $1\frac{1}{2}\% = 1.5\% = 0.015$ (in decimal).

The time is six months, which is $\frac{6}{12}= 0.5$ years.

Step2: Calculate the amount $A$

Substitute $P = 30000$, $r = 0.015$, and $t = 0.5$ into the formula $A = Pe^{rt}$.

$A=30000\times e^{0.015\times0.5}$

First, calculate the exponent: $0.015\times0.5 = 0.0075$

Then, $e^{0.0075}\approx1.007528$ (using a calculator to find the value of $e^{0.0075}$)

Now, calculate $A$: $A = 30000\times1.007528 = 30225.84$

Step3: Calculate the interest earned $I$

The interest earned $I = A - P$.

$I=30225.84 - 30000 = 225.84$

Answer:

The interest earned is $\$225.84$