# Find the quantity Q(t) of the substance left?

1. Jan 3, 2015

### Math10

1. The problem statement, all variables and given/known data
The half-life of a radioactive substance is 3200 years. Find the quantity Q(t) of the substance left at time t>0 if Q(0)=20 g.

2. Relevant equations
dQ/dt=kQ

3. The attempt at a solution
dQ/dt=kQ
dQ/Q=k dt
ln abs(Q)=kt+C
Q=Ce^(kt)
C=20
Q=20e^(kt)
20e^(3200k)=10
e^(3200k)=1/2
Now I'm stucked. The answer is Q=20e^(-(t*ln2)/3200) g.

2. Jan 3, 2015

### Staff: Mentor

Since the quantity Q can't be negative, there's no need for absolute values.
Take the natural log of both sides to solve for k.
There is no such word in English as "stucked." You can say that you're stuck.

3. Jan 3, 2015

### Math10

Okay, thank you!

4. Jan 3, 2015

### Ray Vickson

It is a good idea to get in the habit of doing the following:
(1) Write $Q = e^{-kt}$ if $Q = Q(t)$ is decreasing.
(2) Write $Q = e^{kt}$ if $Q = Q(t)$ is increasing.
By doing that, we always have $k > 0$. This is convenient because we usually want to know the magnitude of $k$ (for example, it might be tabulated in a handbook), and then knowing whether or not you should use $+k$ or $-k$ in the exponent is up to the user. (Of course, your method is OK too, but it will give a negative value of $k$.) This sounds like a minor point, but standard usage in most fields adheres to (1) or (2).