if $1000 is deposited in an account paying 7% annual interest, compounded continuously, how long will it…

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 annual interest rate (in decimal form), and (t) is the time in years.

Given (P=$1000), (A = $1800), and (r=0.07). Substitute these values into the formula: (1800=1000e^{0.07t}).

Step2: Simplify the equation

Divide both sides of the equation by (1000): (\frac{1800}{1000}=e^{0.07t}), so (1.8 = e^{0.07t}).

Step3: Take the natural logarithm of both sides

Using the property (\ln(e^{x})=x), if (y = e^{x}), then (x=\ln(y)).

Take the natural logarithm of both sides: (\ln(1.8)=\ln(e^{0.07t})). Since (\ln(e^{0.07t}) = 0.07t), the equation becomes (\ln(1.8)=0.07t).

Step4: Solve for (t)

We know that (\ln(1.8)\approx0.587787). Then (t=\frac{\ln(1.8)}{0.07}).

Substitute (\ln(1.8)\approx0.587787) into the formula: (t=\frac{0.587787}{0.07}\approx8.40).

Answer:

(8.40)