If you invested $500 into a bank account earning continuously compounded interest and the value after 6 years is $614.99, find the interest rate. Write your answer as a percent rounded to the nearest hundredth.

r = 3.45%

Step-by-step explanation:

You need the formula for continuously compounding interest, which is

A = Pe^ (rt), where

A is the total amount of money in the account,

P is the initial investment

e is a constant that represents continuous growth

r is the interst rate as a decimal (which we will be solving for)

t is how many years the interest was compounded

Plug in the given information

614.99 = 500e^ (r6) (we need to solve for r)

614.99/500 = e^ (6r) (divide both sides by 500, simplify the righ side)

1.22998 = e^ (6r) (simpligy the left side)

When using 'e', we can take the natural log of both sides of the equation and use rules for logarithms to simplify further ...

ln (1.22998) = ln (e^ (6r)

ln (1.222998) = (6r) (ln e) (exponent rule for natural logs lets us bring

down any exponents as multipliers

ln (1.22998) = 6r (ln e = 1, you can verify this on your calculator)

ln (1.22998) / 6 = r (divide both sides by 6 to isolate r)

.034499652 = r (Use a calculator to simplify the left side)

We need to write r as a percent, rounded to the nearest hundredth, so multiply r by 100%, then round

(0.034499652) (100%) = 3.4499652%

Rounding gives 3.45%