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22 December, 03:48

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.

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  1. 22 December, 04:06
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    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%
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