Question

an initial deposit is made in a bank account. find the interest rate, r, if the interest is compounded continuously and no withdrawals or further deposits are made. round to the nearest hundredth of a percent.
initial amount: $3,500: amount in 7 years: $5,400
r = %
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Explanation:

Step1: Write the continuous - compounding formula

The formula for continuous compounding is \(A = Pe^{rt}\), where \(A\) is the final amount, \(P\) is the principal amount, \(r\) is the interest rate, and \(t\) is the time in years.
Given \(P=\$3500\), \(A = \$5400\), and \(t = 7\) years. Substitute these values into the formula: \(5400=3500e^{7r}\).

Step2: Solve for \(e^{7r}\)

Divide both sides of the equation \(5400 = 3500e^{7r}\) by \(3500\):
\(\frac{5400}{3500}=e^{7r}\), which simplifies to \(\frac{54}{35}=e^{7r}\).

Step3: Take the natural logarithm of both sides

Since \(y = e^{x}\) implies \(x=\ln(y)\), for \(e^{7r}=\frac{54}{35}\), we have \(7r=\ln(\frac{54}{35})\).
Calculate \(\ln(\frac{54}{35})\approx\ln(1.5429)\approx0.432\).

Step4: Solve for \(r\)

Divide both sides of the equation \(7r = 0.432\) by \(7\): \(r=\frac{0.432}{7}\approx0.0617\).

Step5: Convert \(r\) to a percentage

Multiply \(r\) by \(100\) to get the percentage: \(r = 0.0617\times100 = 6.17\%\)

Answer:

\(6.17\)