$500 were deposited into an account with a 8.5% interest rate, compounded continuously. how many years was…

$500 were deposited into an account with a 8.5% interest rate, compounded continuously. how many years was it in the bank if the current amount is $3500? t = ? years

$500 were deposited into an account with a 8.5% interest rate, compounded continuously. how many years was it in the bank if the current amount is $3500? t = ? years

Answer

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 (in decimal form), and (t) is the time in years. Given (P=$500), (r = 0.085), and (A=$3500). Substitute these values into the formula: (3500=500e^{0.085t}).

Step2: Simplify the equation

Divide both sides of the equation by (500): (\frac{3500}{500}=e^{0.085t}), so (7 = e^{0.085t}).

Step3: Take the natural logarithm of both sides

Using the property (\ln(e^{x})=x), if (7 = e^{0.085t}), then (\ln(7)=\ln(e^{0.085t})). So (\ln(7)=0.085t).

Step4: Solve for (t)

We know that (\ln(7)\approx1.94591). Then (t=\frac{\ln(7)}{0.085}). Substitute (\ln(7)\approx1.94591) into the formula: (t=\frac{1.94591}{0.085}\approx22.9).

Answer:

(t\approx22.9) years