1. ## Exponential Decay

A substance undergoes radioactive decay. If you start with 3 grams, after 4 seconds you will have 1 gram left.

a) How long will it take to have 2 grams left?
b) How much will you have after 6 seconds?
c) What is the half-life of the substance?

I have asked about these problems on this site multiple times and I am still very confused on how to do these decay and growth problems , any help would be appreciated

2. Amount left = initial amount * e^(rate * time)

1 = 3 * e^ (rate * 4)

Let's solve for the rate:

1/3 = e^(4r)
ln(1/3) = 4r
(1/4)ln(1/3) = r

a) How long will it take to have 2 grams left?
Solve for time in this equation:

2 = 3 e^(rt) // we solved for r above, plug it in and isolate t

b) How much will you have after 6 seconds?
Solve for y:

y = 3 e^(6r) // again we found r above

c) What is the half-life of the substance?
Solve for t:

(3/2) = 3 e^(rt) // once more, we already know r.

These styles of problem have 4 variables: final amount, initial amount, rate, and time. Arrange a problem that plugs in 3 things, and find the 4th.