4.5\nscore: 21/24 answered: 21/24\nquestion 22\nan initial deposit is made in a bank account. find the…

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)