Question

find the principal needed now to get the given amount; that is, find the present value. to get $100 after $1\frac{3}{4}$ years at 8% compounded continuously the present value of $100 is $\square$. (round to the nearest cent as needed.)

Explanation:

Step1: Recall the formula for continuous compounding

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the amount, $P$ is the principal (present value), $r$ is the annual interest rate (in decimal), and $t$ is the time in years. We need to solve for $P$, so we can rearrange the formula to $P=\frac{A}{e^{rt}}$.

Step2: Convert the time and rate to appropriate forms

First, convert the mixed number time $1\frac{3}{4}$ years to an improper fraction or decimal. $1\frac{3}{4}=\frac{7}{4} = 1.75$ years. The interest rate $r = 8\%=0.08$. The amount $A = 100$.

Step3: Substitute the values into the formula

Substitute $A = 100$, $r = 0.08$, and $t = 1.75$ into the formula for $P$:
\[
P=\frac{100}{e^{0.08\times1.75}}
\]

Step4: Calculate the exponent

First, calculate the exponent $rt$: $0.08\times1.75 = 0.14$.

Step5: Calculate the denominator

Now, calculate $e^{0.14}$. Using a calculator, $e^{0.14}\approx1.1503$.

Step6: Calculate the present value

Now, divide $100$ by $e^{0.14}$: $P=\frac{100}{1.1503}\approx86.93$.

Answer:

$\$86.93$