Question

how much would you have to deposit in an account with a 7.5% interest rate, compounded continuously, to have $2500 in your account 8 years later? p = $?

Explanation:

Step1: Recall the continuous - compounding formula

The formula for continuous compounding is \(A = Pe^{rt}\), where \(A\) is the final amount, \(P\) is the principal (initial deposit), \(r\) is the annual interest rate (in decimal form), and \(t\) is the time in years.
We are given \(A=\$2500\), \(r = 0.075\), and \(t = 8\). We need to solve for \(P\).
From \(A = Pe^{rt}\), we can isolate \(P\) by the formula \(P=\frac{A}{e^{rt}}\).

Step2: Substitute the values into the formula

Substitute \(A = 2500\), \(r=0.075\), and \(t = 8\) into \(P=\frac{A}{e^{rt}}\).
We get \(P=\frac{2500}{e^{(0.075\times8)}}\).
First, calculate the exponent: \(0.075\times8=0.6\).
So \(P=\frac{2500}{e^{0.6}}\).
Since \(e^{0.6}\approx1.8221188\).
Then \(P=\frac{2500}{1.8221188}\).

Step3: Calculate the value of \(P\)

\(P=\frac{2500}{1.8221188}\approx1372\).

Answer:

\(1372\)