sophie opened a savings account 5 years ago. the account earns 7% interest, compounded continuously. if the…

Explanation:

Step1: Recall continuous - compounding formula

The formula for continuous - compounding is $A = Pe^{rt}$, where $A$ is the final amount, $P$ is the principal amount (initial deposit), $r$ is the annual interest rate (in decimal form), and $t$ is the time in years. We are given that $A=$100$, $r = 0.07$ (since $7%=0.07$), and $t = 5$ years. We need to solve for $P$.

Step2: Rearrange the formula to solve for $P$

Starting with $A = Pe^{rt}$, we can isolate $P$ by dividing both sides of the equation by $e^{rt}$. So, $P=\frac{A}{e^{rt}}$.

Step3: Substitute the given values into the formula

Substitute $A = 100$, $r=0.07$, and $t = 5$ into the formula $P=\frac{A}{e^{rt}}$. We get $P=\frac{100}{e^{0.07\times5}}=\frac{100}{e^{0.35}}$.

Step4: Calculate the value of $P$

Using a calculator, $e^{0.35}\approx1.419067$, and $\frac{100}{1.419067}\approx70.47$.

Answer:

$70.47$