Question

4.5
score: 21/24 answered: 21/24
question 22
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 3 years: $4,200
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 \(P=\$3500\), \(A = \$4200\), and \(t = 3\) years. Substitute these values into the formula: \(4200=3500e^{3r}\).

Step2: Solve for \(e^{3r}\)

Divide both sides of the equation \(4200 = 3500e^{3r}\) by \(3500\):
\(\frac{4200}{3500}=e^{3r}\).
Simplify \(\frac{4200}{3500}=\frac{6}{5} = 1.2\), so \(1.2=e^{3r}\).

Step3: Take the natural logarithm of both sides

Since \(y = e^{x}\) and \(x=\ln(y)\) are inverse functions, take the natural logarithm of both sides of \(1.2=e^{3r}\).
\(\ln(1.2)=\ln(e^{3r})\).
Using the property \(\ln(e^{x})=x\), we get \(\ln(1.2)=3r\).

Step4: Solve for \(r\)

We know that \(\ln(1.2)\approx0.18232\). Then \(r=\frac{\ln(1.2)}{3}\).
\(r=\frac{0.18232}{3}\approx0.0608\).

Step5: Convert \(r\) to a percentage

To convert \(r\) to a percentage, multiply by \(100\). So \(r = 0.0608\times100 = 6.08\%\).

Answer:

\(6.08\)