Acertain substance decays exponentially according to the function n(t) = n0e−ln(3)t where n(t) is the amount of the substance present after t hours and n0 is the initial amount. how long does it take 27 grams of the substance to decay to 9 grams? simplify your answer completely.

  1 hour

Step-by-step explanation:

Filling in the given information and solving for t, we have ...

  9 = 27·e^(-ln(3)t)

  9/27 = (1/3)^t . . . . . . . . use e^-ln(3) = 1/3

  (1/3)^1 = (1/3)^t . . . . . . . simplify on the left and show an exponent of 1

  1 = t . . . . . . . . . . . . . . . match exponents of the same base

It takes 1 hour for 27 grams to decay to 9 grams.

We have made use of a couple of rules of exponents and logarithms:

  a^(bc) = (a^b)^c

  e^(ln(x)) = x

  -ln(x) = ln(1/x)

step-by-step explanation:

this can be solved first with pythagorean theorem, which tells us to square both legs of the triangle and sum it. this will be equal to the square of the hypotenuse.

the hypotenuse is the side "opposite" the 90 degree angle, which is x.

so we can write using pythagorean theorem,

simplifying:

to further simplify this radical, we need to use the property  

thus